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Academic literature on the topic 'Curvature functionals'

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Published: 14 March 2023

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Journal articles on the topic "Curvature functionals"

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Ivochkina, N. M. "Minimization of functionals generating curvature operators." Journal of Soviet Mathematics 62, no. 3 (November 1992): 2741–46. http://dx.doi.org/10.1007/bf01670999.

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Sheng, Weimin, and Lisheng Wang. "Variational properties of quadratic curvature functionals." Science China Mathematics 62, no. 9 (June 15, 2018): 1765–78. http://dx.doi.org/10.1007/s11425-017-9232-6.

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Brozos‐Vázquez, Miguel, Sandro Caeiro‐Oliveira, and Eduardo García‐Río. "Critical metrics for all quadratic curvature functionals." Bulletin of the London Mathematical Society 53, no. 3 (January 13, 2021): 680–85. http://dx.doi.org/10.1112/blms.12448.

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Kuwert, Ernst, Tobias Lamm, and Yuxiang Li. "Two-dimensional curvature functionals with superquadratic growth." Journal of the European Mathematical Society 17, no. 12 (2015): 3081–111. http://dx.doi.org/10.4171/jems/580.

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Joshi, Pushkar, and Carlo Séquin. "Energy Minimizers for Curvature-Based Surface Functionals." Computer-Aided Design and Applications 4, no. 5 (January 2007): 607–17. http://dx.doi.org/10.1080/16864360.2007.10738495.

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von der Mosel, Heiko. "Nonexistence results for extremals of curvature functionals." Archiv der Mathematik 69, no. 5 (November 1, 1997): 427–34. http://dx.doi.org/10.1007/s000130050141.

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Biondi, Biondo. "Velocity estimation by image-focusing analysis." GEOPHYSICS 75, no. 6 (November 2010): U49—U60. http://dx.doi.org/10.1190/1.3506505.

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Migration velocity can be estimated from seismic data by analyzing, focusing, and defocusing of residual-migrated images. The accuracy of these velocity estimates is limited by the inherent ambiguity between velocity and reflector curvature. However, velocity resolution improves when reflectors with different curvatures are present. Image focusing is measured by evaluating coherency across structural dips, in addition to coherency across aperture/azimuth angles. The inherent ambiguity between velocity and reflector curvature is directly tackled by introducing a curvature correction into the computation of the semblance functional that estimates image coherency. The resulting velocity estimator provides velocity estimates that are (1) unbiased by reflector curvature and (2) consistent with the velocity information that is routinely obtained by measuring coherency over aperture/azimuth angles. Applications to a 2D synthetic prestack data set and a 2D field prestack data set confirm that the proposed method provides consistent and unbiased velocity information. They also suggest that velocity estimates based on the new image-focusing semblance may be more robust and have higher resolution than estimates based on conventional semblance functionals. Applying the proposed method to zero-offset field data recorded in New York Harbor yields a velocity function that is consistent with available geologic information and clearly improves the focusing of the reflectors.
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Sarkar, Prakash. "Quantifying the Cosmic Web using the Shapefinder diagonistic." Proceedings of the International Astronomical Union 11, S308 (June 2014): 250–53. http://dx.doi.org/10.1017/s1743921316009960.

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AbstractOne of the most successful method in quantifying the structures in the Cosmic Web is the Minkowski Functionals. In 3D, there are four minkowski Functionals: Area, Volume, Integrated Mean Curvature and the Integrated Gaussian Curvature. For defining the Minkowski Functionals one should define a surface. We have developed a method based on Marching cube 33 algorithm to generate a surface from a discrete data sets. Next we calculate the Minkowski Functionals and Shapefinder from the triangulated polyhedral surface. Applying this methodology to different data sets , we obtain interesting results related to geometry, morphology and topology of the large scale structure
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Pulemotov, Artem. "Maxima of Curvature Functionals and the Prescribed Ricci Curvature Problem on Homogeneous Spaces." Journal of Geometric Analysis 30, no. 1 (March 6, 2019): 987–1010. http://dx.doi.org/10.1007/s12220-019-00175-6.

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Fierro, F., R. Goglione, and M. Paolini. "Finite element minimization of curvature functionals with anisotropy." Calcolo 31, no. 3-4 (September 1994): 191–210. http://dx.doi.org/10.1007/bf02575878.

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Dissertations / Theses on the topic "Curvature functionals"

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Mondino, Andrea. "The Willmore functional and other L^p curvature functionals in Riemannian manifolds." Doctoral thesis, SISSA, 2011. http://hdl.handle.net/20.500.11767/4840.

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Using techniques both of non linear analysis and geometric measure theory, we prove existence of minimizers and more generally of critical points for the Willmore functional and other $L^p$ curvature functionals for immersions in Riemannian manifolds. More precisely, given a $3$-dimensional Riemannian manifold $(M,g)$ and an immersion of a sphere $f:\Sp^2 \hookrightarrow (M,g)$ we study the following problems. 1) The Conformal Willmore functional in a perturbative setting: consider $(M,g)=(\Rtre,\eu+\epsilon h)$ the euclidean $3$-space endowed with a perturbed metric ($h=h_{\mu\nu}$ is a smooth field of symmetric bilinear forms); we prove, under assumptions on the trace free Ricci tensor and asymptotic flatness, existence of critical points for the Conformal Willmore functional $I(f):=\frac{1}{2}\int |A^\circ|^2 $ (where $A^\circ:=A-\frac{1}{2}H$ is the trace free second fundamental form). The functional is conformally invariant in curved spaces. We also establish a non existence result in general Riemannian manifolds. The technique is perturbative and relies on a Lyapunov-Schmidt reduction. \\ 2) The Willmore functional in a semi-perturbative setting: consider $(M,g)=(\Rtre, \eu+h)$ where $h=h_{\mu\nu}$ is a $C^{\infty}_0(\Rtre)$ field of symmetric bilinear forms with compact support and small $C^1$ norm. Under a general assumption on the scalar curvature we prove existence of a smooth immersion of $\Sp^2$ minimizing the Willmore functional $W(f):=\frac{1}{4} \int |H|^2$ (where $H$ is the mean curvature). The technique is more global and relies on the direct method in the calculus of variations. \\ 3) The functionals $E:=\frac{1}{2} \int |A|^2 $ and $W_1:=\int\left( \frac{|H|^2}{4}+1 \right)$ in compact ambient manifolds: consider $(M,g)$ a $3$-dimensional compact Riemannian manifold. We prove, under global conditions on the curvature of $(M,g)$, existence and regularity of an immersion of a sphere minimizing the functionals $E$ or $W_1$. The technique is global, uses geometric measure theory and regularity theory for higher order PDEs. \\ 4) The functionals $E_1:=\int \left( \frac{|A|^2}{2} +1 \right) $ and $W_1:=\int\left( \frac{|H|^2}{4}+1 \right)$ in noncompact ambient manifolds: consider $(M,g)$ a $3$-dimensional asymptotically euclidean non compact Riemannian $3$-manifold. We prove, under general conditions on the curvature of $(M,g)$, existence and regularity of an immersion of a sphere minimizing the functionals $E_1$ or $W_1$. The technique relies on the direct method in the calculus of variations. \\ 5) The supercritical functionals $\int |H|^p$ and $\int |A|^p$ in arbitrary dimension and codimension: consider $(N,g)$ a compact $n$-dimensional Riemannian manifold possibly with boundary. For any $2\leq mm$, defined on the $m$-dimensional submanifolds of $N$. We prove, under assumptions on $(N,g)$, existence and partial regularity of a minimizer of such functionals in the framework of varifold theory. During the arguments we prove some new monotonicity formulas and new Isoperimetric Inequalities which are interesting by themselves.
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PARRILLO, ANTONELLA. "Analytical and computational study of curvature depending functionals in image segmentation." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2008. http://hdl.handle.net/2108/689.

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In questa tesi sono stati trattati problemi di segmentazione dell'immagine mediante strumenti di analisi variazionale. Ho studiato due funzionali contenenti integrali di funzioni dipendenti dalla curvatura degli elementi di una famiglia di curve $C$ approssimante i contorni di una data immagine, la lunghezza di esse e il numero dei loro punti finali. Per uno dei due funzionali ho calcolato il sistema delle equazioni di Eulero e, usando uno schema iterativo basato sulle differenze finite, ho effettuato esperimenti al computer su alcune immagini.
In the present thesis we study variational problems for image segmentation. We consider two specific classes of functionals which contain the integral of a function of curvature along the unknown set of curves $C$, the length of such curves and the counting measure of the set of theirs endpoints. For the second functionals we derive the system of Euler equations, we design an iterative numerical scheme based on finite differences for the solution of the Euler equations, and we discuss the outcome of some computer experiments on simulated images.
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Winklmann, Sven. "Krümmungsabschätzungen für stabile Extremalen parametrischer Funktionale / Curvature estimates for stable extremals of parametric functionals." Gerhard-Mercator-Universitaet Duisburg, 2004. http://www.ub.uni-duisburg.de/ETD-db/theses/available/duett-03192004-115454/.

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We consider immersed hypersurfaces in euclidean $R^{n+1}$ which are stable with respect to an elliptic parametric functional with integrand $F=F(N)$ depending on normal directions only. We prove an integral curvature estimate provided that $F$ is sufficiently close to the area integrand, extending the classical curvature estimate of Schoen, Simon and Yau for stable minimal hypersurfaces in $R^{n+1}$. As a crucial point of our analysis we derive a generalized Simons inequality for the laplacian of the length of a weighted second fundamental form with respect to an abstract metric associated with $F$. Using Moser's iteration technique we finally prove a pointwise curvature estimate for $n leq 5$. As an application we obtain a new Bernstein result for complete stable hypersurfaces of dimension $n leq 5$.
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Käfer, Bastian [Verfasser], der Mosel Heiko Akademischer Betreuer] von, Alfred [Akademischer Betreuer] [Wagner, and Pawel [Akademischer Betreuer] Strzelecki. "Scale-invariant geometric curvature functionals, and characterization of Lipschitz- and $C^1$-submanifolds / Bastian Käfer ; Heiko von der Mosel, Alfred Wagner, Pawel Strzelecki." Aachen : Universitätsbibliothek der RWTH Aachen, 2021. http://d-nb.info/1239566719/34.

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Silva, Adam Oliveira da. "Rigidez de métricas críticas para funcionais riemannianos." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/25969.

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SILVA, Adam Oliveira da. Rigidez de métricas críticas para funcionais riemannianos. 2017. 78 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.
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The aim of this work is to study metrics that are critical points for some Riemannian functionals. In the first part, we investigate critical metrics for functionals which are quadratic in the curvature on closed Riemannian manifolds. It is known that space form metrics are critical points for these functionals, denoted by F t,s (g). Moreover, when s = 0, always Einstein metrics are critical to F t (g). We proved that under some conditions the converse is true. For instance, among others results, we prove that if n ≥ 5 and g is a Bach-flat critical metric to F −n/4(n−1) , with second elementary symmetric function of the Schouten tensor σ 2 (A) > 0, then g should be Einstein. Furthermore, we show that a locally conformally flat critical metric with some additional conditions are space form metrics. In the second part, we study the critical metrics to volume functional on compact Riemannian manifolds with connected smooth boundary. We call such critical points of Miao-Tam critical metrics due to the variational study making by Miao and Tam (2009). In this work, we show that the geodesics balls in space forms Rn , Sn and Hn have the maximum possible boundary volume among Miao-Tam critical metrics with connected boundary provided that the boundary be an Einstein manifold. In the same spirit, we also extend a rigidity theorem due to Boucher et al. (1984) and Shen (1997) to n-dimensional static metrics with positive constant scalar curvature, which give us another way to get a partial answer to the Cosmic no-hair conjecture already obtained by Chrusciel (2003).
Este trabalho tem como principal objetivo estudar métricas que são pontos críticos de alguns funcionais Riemannianos. Na primeira parte, investigaremos métricas críticas de funcionais que são quadráticos na curvatura sobre variedades Riemannianas fechadas. É de conhecimento que métricas tipo formas espaciais são pontos críticos para tais funcionais, denotados aqui por F t,s (g). Além disso, no caso s = 0, métricas de Einstein são sempre críticas para F t (g). Provamos que sob algumas condições, a recíproca destes fatos são verdadeiras. Por exemplo, dentre outros resultados, provamos que se n ≥ 5 e g é uma métrica Bach-flat crìtica para F−n/4(n−1) com segunda função simétrica elementar do tensor de Schouten σ 2 (A) > 0, então g tem que ser métrica de Einstein. Ademais, mostramos que uma métrica crítica localmente conformemente plana, com algumas hipóteses adicionais, tem que ser tipo forma espacial. Na segunda parte, estudamos as métricas críticas do funcional volume sobre variedades Riemannianas compactas com bordo suave conexo. Chamamos tais pontos críticos de métricas críticas de Miao-Tam, devido ao estudo variacional feito por Miao e Tam (2009). Neste trabalho provamos que as bolas geodésicas das formas espaciais Rn , S n e H n possuem o valor máximo para o volume do bordo dentre todas as métricas críticas de Miao-Tam com bordo conexo, desde que o bordo seja uma variedade de Einstein. No mesmo sentido, também estendemos um teorema de rigidez devido à Boucher et al. (1984) e Shen (1997) para métricas estáticas de dimensão n e com curvatura escalar constante positiva, o qual nos fornece outra maneira para obter uma resposta parcial para a Cosmic no-hair conjecture já obtida por Chrusciel (2003).
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Guo, Li. "Shape blending using discrete curvature-variation functional /." View abstract or full-text, 2005. http://library.ust.hk/cgi/db/thesis.pl?IEEM%202005%20GUO.

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Dalphin, Jérémy. "Étude de fonctionnelles géométriques dépendant de la courbure par des méthodes d'optimisation de formes. Applications aux fonctionnelles de Willmore et Canham-Helfrich." Thesis, Université de Lorraine, 2014. http://www.theses.fr/2014LORR0167/document.

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En biologie, lorsqu'une quantité importante de phospholipides est insérée dans un milieu aqueux, ceux-Ci s'assemblent alors par paires pour former une bicouche, plus communément appelée vésicule. En 1973, Helfrich a proposé un modèle simple pour décrire la forme prise par une vésicule. Imposant la surface de la bicouche et le volume de fluide qu'elle contient, leur forme minimise une énergie élastique faisant intervenir des quantités géométriques comme la courbure, ainsi qu'une courbure spontanée mesurant l'asymétrie entre les deux couches. Les globules rouges sont des exemples de vésicules sur lesquels sont fixés un réseau de protéines jouant le rôle de squelette au sein de la membrane. Un des principaux travaux de la thèse fut d'introduire et étudier une condition de boule uniforme, notamment pour modéliser l'effet du squelette. Dans un premier temps, on cherche à minimiser l'énergie de Helfrich sans contrainte puis sous contrainte d'aire. Le cas d'une courbure spontanée nulle est connu sous le nom d'énergie de Willmore. Comme la sphère est un minimiseur global de l'énergie de Willmore, c'est un bon candidat pour être un minimiseur de l'énergie de Helfrich parmi les surfaces d'aire fixée. Notre première contribution dans cette thèse a été d'étudier son optimalité. On montre qu'en dehors d'un certain intervalle de paramètres, la sphère n'est plus un minimum global, ni même un minimum local. Par contre, elle est toujours un point critique. Ensuite, dans le cas de membranes à courbure spontanée négative, on se demande si la minimisation de l'énergie de Helfrich sous contrainte d'aire peut être effectuée en minimisant individuellement chaque terme. Cela nous conduit à minimiser la courbure moyenne totale sous contrainte d'aire et à déterminer si la sphère est la solution de ce problème. On montre que c'est le cas dans la classe des surfaces axisymétriques axiconvexes mais que ce n'est pas vrai en général.Enfin, lorsqu'une contrainte d'aire et de volume sont considérées simultanément, le minimiseur ne peut pas être une sphère qui n'est alors plus admissible. En utilisant le point de vue de l'optimisation de formes, la troisième et plus importante contribution de cette thèse est d'introduire une classe plus raisonnable de surfaces, pour laquelle l'existence d'un minimiseur suffisamment régulier est assurée pour des fonctionnelles et des contraintes générales faisant intervenir les propriétés d'ordre un et deux des surfaces. En s'inspirant de ce que fit Chenais en 1975 quand elle a considéré la propriété de cône uniforme, on considère les surfaces satisfaisant une condition de boule uniforme. On étudie d'abord des fonctionnelles purement géométriques puis nous autorisons la dépendance à travers la solution de problèmes aux limites elliptiques d'ordre deux posés sur le domaine intérieur à la surface
In biology, when a large amount of phospholipids is inserted in aqueous media, they immediatly gather in pairs to form bilayers also called vesicles. In 1973, Helfrich suggested a simple model to characterize the shapes of vesicles. Imposing the area of the bilayer and the volume of fluid it contains, their shape is minimizing a free-Bending energy involving geometric quantities like curvature, and also a spontanuous curvature measuring the asymmetry between the two layers. Red blood cells are typical examples of vesicles on which is fixed a network of proteins playing the role of a skeleton inside the membrane. One of the main work of this thesis is to introduce and study a uniform ball condition, in particular to model the effects of the skeleton. First, we minimize the Helfrich energy without constraint then with an area constraint. The case of zero spontaneous curvature is known as the Willmore energy. Since the sphere is the global minimizer of the Willmore energy, it is a good candidate to be a minimizer of the Helfrich energy among surfaces of prescribed area. Our first main contribution in this thesis was to study its optimality. We show that apart from a specific interval of parameters, the sphere is no more a global minimizer, neither a local minimizer. However, it is always a critical point. Then, in the specific case of membranes with negative spontaneous curvature, one can wonder whether the minimization of the Helfrich energy with an area constraint can be done by minimizing individually each term. This leads us to minimize total mean curvature with prescribed area and to determine if the sphere is a solution to this problem. We show that it is the case in the class of axisymmetric axiconvex surfaces but that it does not hold true in the general case. Finally, considering both area and volume constraints, the minimizer cannot be the sphere, which is no more admissible. Using the shape optimization point of view, the third main and most important contribution of this thesis is to introduce a more reasonable class of surfaces, in which the existence of an enough regular minimizer is ensured for general functionals and constraints involving the first- and second-Order geometric properties of surfaces. Inspired by what Chenais did in 1975 when she considered the uniform cone property, we consider surfaces satisfying a uniform ball condition. We first study purely geometric functionals then we allow a dependence through the solution of some second-Order elliptic boundary value problems posed on the inner domain enclosed by the shape
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ALESSANDRONI, ROBERTA. "Evolution of hypersurfaces by curvature functions." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2008. http://hdl.handle.net/2108/661.

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Consideriamo un'ipersuperficie liscia di ℝⁿ⁺¹, con n≥2, e la sua evoluzione secondo una classe di flussi geometrici. La velocità di questi flussi ha direzione normale alla superficie e il modulo è una funzione simmetrica delle curvature principali. Inizialmente mostriamo alcune proprietà generali di questi flussi e calcoliamo l'equazione di evoluzione per una generica funzione omogenea delle curvature principali. In particolare applichiamo il flusso con velocità S=(H/(logH)), dove H è la curvatura media a meno di una costante, ad una superficie con curvatura media positiva per ottenere delle stime di convessità. Usando solamente il principio del massimo dimostriamo che, su un limite di riscalamenti delle superfici che si evolvono vicino alla singolarità, la parte negativa della curvatura scalare tende a zero. La parte successiva è dedicata allo studio di un'ipersuperficie convessa che si evolve secondo potenze della curvatura scalare: S=R^{p}, con p>1/2. Si dimostra che se la superficie iniziale soddisfa delle stime di "pinching" sulle curvature principali allora si contrae ad un punto in tempo finito e la forma delle superfici che si evolvono approssima sempre più quella di una sfera. In questo caso il grado di omogeneità, strettamente maggiore di uno, permette di concludere la dimostrazione della convergenza ad un "punto rotondo" tramite il solo principio del massimo, evitando l'uso di stime integrali. Viene anche costruito un esempio di superficie convessa che forma una singolarità di tipo "neck pinching". Infine studiamo il caso di un grafico intero su ℝⁿ con crescita al più lineare all'infinito e mostriamo che un grafico che si evolve secondo un qualsiasi flusso nella classe considerata rimane un grafico. Inoltre dimostriamo un risultato di esistenza per tempi lunghi per i flussi con velocità S=R^{p} con p≥1/2 e descriviamo delle soluzioni esplicite per grafici a simmetria di rotazione.
We consider a smooth n-dimensional hypersurface of ℝⁿ⁺¹, with n≥2, and its evolution by a class of geometric flows. The speed of these flows has normal direction with respect to the surface and its modulus S is a symmetric function of the principal curvatures. We show some general properties of these flows and compute the evolution equation for any homogeneous function of principal curvatures. Then we apply the flow with speed S=(H/(logH)), where H is the mean curvature plus a constant, to a mean convex surface to prove some convexity estimates. Using only the maximum principle we prove that the negative part of the scalar curvature tends to zero on a limit of rescalings of the evolving surfaces near a singularity. The following part is dedicated to the study of a convex initial manifold moving by powers of scalar curvature: S=R^{p}, with p>1/2. We show that if the initial surface satisfies a pinching estimate on the principal curvatures then it shrinks to a point in finite time and the shape of the evolving surfaces approaches the one of a sphere. Since the homogeneity degree of this speed is strictly greater than one, the convergence to a "round point" can be proved using just the maximum principle, avoiding the integral estimates. Then we also construct an example of a non convex surface forming a neck pinching singularity. Finally we study the case of an entire graph over ℝⁿ with at most linear growth at infinity. We show that a graph evolving by any flow in the considered class remains a graph. Moreover we prove a long time existence result for flows where the speed is S=R^{p} with p≥1/2 and describe some explicit solutions in the rotationally symmetric case.
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Evangelista, Israel de Sousa. "Compact almost Ricci soliton, critical metrics of the total scalar curvature functional and p-fundamental tone estimates." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/23920.

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EVANGELISTA, I. S. Compact almost Ricci soliton, critical metrics of the total scalar curvature functional and p-fundamental tone estimates. 2017. 75 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.
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The present thesis is divided in three different parts. The aim of the first part is to prove that a compact almost Ricci soliton with null Cotton tensor is isometric to a standard sphere provided one of the following conditions associated to the Schouten tensor holds: the second symmetric function is constant and positive; two consecutive symmetric functions are non null multiple or some symmetric function is constant and the quoted tensor is positive. The aim of the second part is to study the critical metrics of the total scalar curvature funcional on compact manifolds with constant scalar curvature and unit volume, for simplicity, CPE metrics. It has been conjectured that every CPE metric must be Einstein. We prove that the Conjecture is true for CPE metrics under a suitable integral condition and we also prove that it suffices the metric to be conformal to an Einstein metric. In the third part we estimate the p-fundamental tone of submanifolds in a Cartan-Hadamard manifold. First we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. Moreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension one C 2-foliations of open subsets Ω of Riemannian manifolds M and obtain lower bounds estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of Ω.
A presente tese está dividida em três partes diferentes. O objetivo da primeira parte é provar que um quase soliton de Ricci compacto com tensor de Cotton nulo é isométrico a uma esfera canônica desde que uma das seguintes condições associadas ao tensor de Schouten seja válida: a segunda função simétrica é constante e positiva; duas funções simétricas consecutivas são múltiplas, não nulas, ou alguma função simétrica é constante e o tensor de Schouten é positivo. O objetivo da segunda parte é estudar as métricas críticas do funcional curvatura escalar total em variedades compactas com curvatura escalar constante e volume unitário, por simplicidade, métricas CPE. Foi conjecturado que toda métrica CPE deve ser Einstein. Prova-se que a conjectura é verdadeira para as métricas CPE sob uma condição integral adequada e também se prova que é suficiente que a métrica seja conforme a uma métrica Einstein. Na terceira parte, estima-se o p-tom fundamental de subvariedades em uma variedade tipo Cartan-Hadamard. Primeiramente, obtém-se estimativas por baixo para o p-tom fundamental de bolas geodésicas e em subvariedades com curvatura média limitada. Além disso, obtém-se estimativas do p-tom fundamental de subvariedades mínimas com certas condições sobre a norma da segunda forma fundamental. Por fim, estudam-se folheações de classe C 2 transversalmente orientadas de codimensão 1 de subconjuntos abertos Ω de variedades riemannianas M e obtêm-se estimativas por baixo para o ínfimo da curvatura média das folhas em termos do p-tom fundamental de Ω.
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Buckland, John A. (John Anthony) 1978. "Mean curvature flow with free boundary on smooth hypersurfaces." Monash University, School of Mathematical Sciences, 2003. http://arrow.monash.edu.au/hdl/1959.1/5809.

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Books on the topic "Curvature functionals"

1

Metrics of positive scalar curvature and generalised Morse functions. Providence, R.I: American Mathematical Society, 2011.

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Yang, Kichoon. Complete Minimal Surfaces of Finite Total Curvature. Dordrecht: Springer Netherlands, 1994.

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Wu, K. Chauncey. Free vibration of hexagonal panels simply supported at discrete points. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1991.

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Houdre, Christian, and Christian Houdré. Concentration, functional inequalities, and isoperimetry: International workshop, October 29-November 1, 2009, Florida Atlantic University, Boca Raton, Florida. Providence, R.I: American Mathematical Society, 2011.

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Li, Weiping, and Shihshu Walter Wei. Geometry and topology of submanifolds and currents: 2013 Midwest Geometry Conference, October 19, 2013, Oklahoma State University, Stillwater, Oklahoma : 2012 Midwest Geometry Conference, May 12-13, 2012, University of Oklahoma, Norman, Oklahoma. Providence, Rhode Island: American Mathematical Society, 2015.

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Wentworth, Richard A., Duong H. Phong, Paul M. N. Feehan, Jian Song, and Ben Weinkove. Analysis, complex geometry, and mathematical physics: In honor of Duong H. Phong : May 7-11, 2013, Columbia University, New York, New York. Providence, Rhode Island: American Mathematical Society, 2015.

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Tretkoff, Paula. Riemann Surfaces, Coverings, and Hypergeometric Functions. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691144771.003.0003.

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This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and notes that an automorphism of a Riemann surface is a biholomorphic map of the Riemann surface onto itself. It then describes a Riemannian metric and the Gauss-Bonnet theorem, which can be interpreted as a relation between the Gaussian curvature of a compact Riemann surface X and its Euler characteristic. It also examines the behavior of the Euler number under finite covering, along with finite subgroups of the group of fractional linear transformations PSL(2, C). Finally, it presents some basic facts about the classical Gauss hypergeometric functions of one complex variable, triangle groups acting discontinuously on one of the simply connected Riemann surfaces, and the hypergeometric monodromy group.
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Briggs, Carey. General Expressions for the 5-Dimensional Riemann-Christoffel, Ricci, and Einstein Curvature Tensors and Riemann Curvature Scalar Allowing for Non-Vanishing Torsion and Arbitrary Functional Dependence on the Fifth Dimension. Lulu Press, Inc., 2021.

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Free vibration of hexagonal panels simply supported at discrete points. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1991.

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Rajeev, S. G. Fluid Mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.001.0001.

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Starting with a review of vector fields and their integral curves, the book presents the basic equations of the subject: Euler and Navier–Stokes. Some solutions are studied next: ideal flows using conformal transformations, viscous flows such as Couette and Stokes flow around a sphere, shocks in the Burgers equation. Prandtl’s boundary layer theory and the Blasius solution are presented. Rayleigh–Taylor instability is studied in analogy with the inverted pendulum, with a digression on Kapitza’s stabilization. The possibility of transients in a linearly stable system with a non-normal operator is studied using an example by Trefethen et al. The integrable models (KdV, Hasimoto’s vortex soliton) and their hamiltonian formalism are studied. Delving into deeper mathematics, geodesics on Lie groups are studied: first using the Lie algebra and then using Milnor’s approach to the curvature of the Lie group. Arnold’s deep idea that Euler’s equations are the geodesic equations on the diffeomorphism group is then explained and its curvature calculated. The next three chapters are an introduction to numerical methods: spectral methods based on Chebychev functions for ODEs, their application by Orszag to solve the Orr–Sommerfeld equation, finite difference methods for elementary PDEs, the Magnus formula and its application to geometric integrators for ODEs. Two appendices give an introduction to dynamical systems: Arnold’s cat map, homoclinic points, Smale’s horse shoe, Hausdorff dimension of the invariant set, Aref ’s example of chaotic advection. The last appendix introduces renormalization: Ising model on a Cayley tree and Feigenbaum’s theory of period doubling.
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Book chapters on the topic "Curvature functionals"

1

Viaclovsky, Jeff. "Critical metrics for Riemannian curvature functionals." In Geometric Analysis, 195–274. Providence, Rhode Island: American Mathematical Society, 2016. http://dx.doi.org/10.1090/pcms/022/05.

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Blair, David E. "Curvature Functionals on Spaces of Associated Metrics." In Riemannian Geometry of Contact and Symplectic Manifolds, 157–75. Boston, MA: Birkhäuser Boston, 2002. http://dx.doi.org/10.1007/978-1-4757-3604-5_10.

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Blair, David E. "Curvature Functionals on Spaces of Associated Metrics." In Riemannian Geometry of Contact and Symplectic Manifolds, 195–218. Boston: Birkhäuser Boston, 2010. http://dx.doi.org/10.1007/978-0-8176-4959-3_10.

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Nitsche, Johannes C. C. "Periodic Surfaces That are Extremal for Energy Functionals Containing Curvature Functions." In Statistical Thermodynamics and Differential Geometry of Microstructured Materials, 69–98. New York, NY: Springer New York, 1993. http://dx.doi.org/10.1007/978-1-4613-8324-6_6.

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Dineen, Seán. "Curvature." In Functions of Two Variables, 103–14. Boston, MA: Springer US, 1995. http://dx.doi.org/10.1007/978-1-4899-3250-1_14.

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Jost, Jürgen. "Convex functions and centers of mass." In Nonpositive Curvature: Geometric and Analytic Aspects, 61–68. Basel: Birkhäuser Basel, 1997. http://dx.doi.org/10.1007/978-3-0348-8918-6_3.

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Sunada, Toshikazu. "L-functions in geometry and some applications." In Curvature and Topology of Riemannian Manifolds, 266–84. Berlin, Heidelberg: Springer Berlin Heidelberg, 1986. http://dx.doi.org/10.1007/bfb0075662.

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Shiohama, Katsuhiro. "Critical points of Busemann functions on complete open surfaces." In Curvature and Topology of Riemannian Manifolds, 254–65. Berlin, Heidelberg: Springer Berlin Heidelberg, 1986. http://dx.doi.org/10.1007/bfb0075661.

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Alexander, Stephanie B., and William A. Karr. "Space-Time Convex Functions and Sectional Curvature." In Lorentzian Geometry and Related Topics, 13–26. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-66290-9_2.

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Kenmotsu, Katsuei. "Harmonic Functions and Parallel Mean Curvature Surfaces." In Springer Proceedings in Mathematics & Statistics, 13–19. Tokyo: Springer Japan, 2014. http://dx.doi.org/10.1007/978-4-431-55215-4_2.

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Conference papers on the topic "Curvature functionals"

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NETERENKO, V. V., A. FEOLI, and G. SCARPETTA. "FUNCTIONALS LINEAR IN CURVATURE AND STATISTICS OF HELICAL PROTEINS." In Proceedings of the International Conference. WORLD SCIENTIFIC, 2004. http://dx.doi.org/10.1142/9789812702883_0047.

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Rajidi, Shashidhar Reddy, Abhay Gupta, and Satyajit Panda. "Supersonic Aerodynamic Instability Characteristics of Bidirectional Porous Functionally Graded Panel." In ASME Turbo Expo 2021: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/gt2021-59346.

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Abstract The objective to alleviate the detrimental effects of supersonic flutter of aerospace structures necessitates the development of advanced composite materials. Porous functionally graded materials are viable alternatives to replace the metal/alloys used for critical components. The present work investigates the supersonic flutter characteristics of hinged-hinged panel for the porosity grading across the thickness and/or along the streamwise direction. Also, the possibility to alleviate the detrimental effects is investigated through the study of influence of streamwise and spanwise curvatures. A geometrically nonlinear finite element model of panel is derived using first-order shear deformation theory while the aerodynamic pressure on panel is accounted using the first-order piston theory. The results revealed that symmetric distribution of porosity with minimum porosity at the midspan and maximum porosity at core displays the better performance. Porosity and streamwise curvature reduces critical aerodynamic pressure and enhances flutter amplitude. For higher streamwise curvatures/porosity, panel undergoes snap-through buckling resulting in complex vibrations. Whereas the spanwise curvature substantially enhance the critical dynamic pressure thereby eliminates complex oscillations and snap-through. But moderately increases the flutter amplitude and frequency beyond its critical aerodynamic pressure. At higher spanwise curvatures, the effectiveness of bidirectional grading decreases making its through-thickness grading as dominant.
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Knippenberg, Christopher H., Oliver J. Myers, and Christopher Nelon. "Functional Description for Thick Bistable Carbon Fiber Laminates With Rayleigh-Ritz, Abaqus, and Experiments." In ASME 2020 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/smasis2020-2293.

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Abstract Composite laminates constructed in an asymmetric layup orientation of [0i, 90i], i > 0, exhibit two stable equilibrium positions and may be actuated to snap from a primary cure shape to an inversely related secondary stable shape. This study aims to aid in developing a comprehensive description of thick bistable laminates, whose increased thickness risks the loss of bistability, through previously established analytical approaches and verification via experimentation. The principle of minimum potential energy is applied to two materials and analyzed using the Rayleigh-Ritz minimization technique to determine the cure shapes of carbon fiber reinforced polymer laminates composed of AS4/8552 and TR50S-12k carbon fibers. These materials were modeled to act as square thick bistable laminated composites with sidelengths up to 0.914m. Visualizations of the out-of-plane displacements are shown with a description of the Rayleigh-Ritz analysis. Additionally, a finite element model (FEM) created in Abaqus CAE 6.14 and experiments using DA409/G35 and TR50S-12K/NP301 prepreg were used to further describe and develop the fundamental description for thick bistable laminates in terms of loss of bistability, actuation load, and principle shape. The analytical model is an extension of Hyer’s (2002) and Mattioni’s (2009) work applied to thick bistable laminates where the primary assumption was the x-axis curvature equaled the negative y-axis curvature for the primary and secondary stable positions, respectively. This assumption leads to the already cemented conclusion that bistable laminates, once cured, take on one of two inversely related paraboloid shapes. FEA simulations contradicted this by showing an average 11% difference in curvature magnitude for the aforementioned shapes. Furthermore, fourth order polynomials were used to describe the curvature along the axes, differing from the previously used Menger curvatures, (three-point approximation). Bifurcation plots using peak deflections and average curvature generated from FEA simulations clearly showed bistability existed to approximately 50 plies; however, the energy landscape plots indicated a significant degradation of bistability starting at 36 plies. Experimentation was performed on a test stand mimicking the same boundary conditions used in FEA while applying a central out-of-plane load. Experimental observations showed decreased peak displacements of stable cure shapes. Observations also indicated that the x-axis curvature had a significant difference in magnitude compared to the negative y-axis curvature. However, the existence of bistability agreed with FEA energy landscape plots, with clear “snaps” ending at thicknesses of 28–36 plies. Moreover, actuation force was found to correlate well with FEA simulations. Differences in the critical point can be attributed to the combination of material property differences for DA409 and TR50S-12K, failure to capture polymer relaxation, limitations of the experimental setup, and hand layup fabrication errors. Lastly, this paper adds viability of thicker laminates for use in macroscale applications where shape morphing or shape-retention attributes are a necessary constraint, although only where low loads are expected.
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Yoshida, Norimasa, and Takafumi Saito. "Planar Curves-based on Explicit Bézier Curvature Functions." In CAD'19. CAD Solutions LLC, 2019. http://dx.doi.org/10.14733/cadconfp.2019.323-327.

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BRITO, F., H. L. LIU, U. SIMON, and C. P. WANG. "HYPERSURFACES IN SPACE FORMS WITH SOME CONSTANT CURVATURE FUNCTIONS." In Geometry and Topology of Submanifolds IX. WORLD SCIENTIFIC, 1999. http://dx.doi.org/10.1142/9789812817976_0006.

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6

Lehky, Sidney R., and Terrence J. Sejnowski. "Extracting 3-D curvatures from images of surfaces using a neural network." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1987. http://dx.doi.org/10.1364/oam.1987.mh3.

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A network which extracts principal curvatures and orientations from images of simple surfaces using shading information was constructed with the backpropagation learning algorithm.1 The surfaces were elliptic paraboloids with a parabolic cross section in depth, an elliptical cross section in the fronto-parallel plane, and a Lambertian reflectance function. The network finds the principal curvatures and directions at the centers of the surfaces over a wide range of values for those parameters, independent of illumination direction and the location of the center. Input is mediated by convolving the image with a hexagonal array of input units with overlapping circularly symmetric Laplacian receptive fields. Output is represented in a distributed fashion in the joint activities of a population of units whose sensitivities are 2-D Gaussian functions in a curvature-orientation parameter space. During learning, a variety of oriented and nonoriented patterns form among the inhibitory and excitatory synaptic weights associated with the hidden units, which are located between the input units and output units in the threelayer network.
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Torselletti, Enrico, Luigino Vitali, Roberto Bruschi, Erik Levold, and Leif Collberg. "Submarine Pipeline Installation Joint Industry Project: Global Response Analysis of Pipelines During S-Laying." In 25th International Conference on Offshore Mechanics and Arctic Engineering. ASMEDC, 2006. http://dx.doi.org/10.1115/omae2006-92377.

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The development of deep water gas fields using trunklines to carry the gas to the markets is sometime limited by the feasibility/economics of the construction phase. In particular there is market for using S-lay vessel in water depth larger than 1000m. The S-lay feasibility depends on the applicable tension at the tensioner which is a function of water depth, stinger length and stinger curvature (for given stinger length by its curvature). This means that, without major vessel up-grading and to avoid too long stingers that are prone to damages caused by environmental loads, the application of larger stinger curvatures than allowed by current regulations/state of the art, is needed. The work presented in this paper is a result of the project “Development of a Design Guideline for Submarine Pipeline Installation” sponsored by STATOIL and HYDRO. The technical activities are performed in co-operation by DNV, STATOIL and SNAMPROGETTI. This paper presents the results of the analysed S-lay scenarios in relation to extended laying ability of medium to large diameter pipelines in order to define the statistical distribution of the relevant load effects, i.e. bending moment and longitudinal strain as per static/functional, dynamic/total, and environmental load effects. The results show that load effects (longitudinal applied strain and bending moment) are strongly influenced by the static setting (applied stinger curvature and axial force at the tensioner in combination with local roller reaction over the stinger). The load effect distributions are the basis for the development of design criteria/safety factors which fulfil a predefined target safety level.
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Yoshida, Norimasa, and Takafumi Saito. "Intrinsically Defined Planar Curves based on Explicit B-spline Curvature Functions." In CAD'21. CAD Solutions LLC, 2021. http://dx.doi.org/10.14733/cadconfp.2021.51-55.

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Liu, Yajing, Edwin K. P. Chong, and Ali Pezeshki. "Extending Polymatroid Set Functions With Curvature and Bounding the Greedy Strategy." In 2018 IEEE Statistical Signal Processing Workshop (SSP). IEEE, 2018. http://dx.doi.org/10.1109/ssp.2018.8450732.

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Yue, Hong-Hao, Xiao-Ying Gao, Bing-Yin Ren, and Horn-Sen Tzou. "Spatial Exact Actuation of Flexible Deep Double-Curvature Shells." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34993.

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Deep double-curvature shells are commonly used as key components in many advanced aerospace structures and mechanical systems, e.g., nozzles, injectors, horns, rocket fairings. Spatially distributed micro-actuation of a laminated flexible deep double curvature shell is investigated and its control effectiveness is evaluated in this study. Dynamic equations of the smart double curvature shell system are presented and modal control forces of spatial segmented piezoelectric actuators are carried out based on a new set of assumed mode shape functions with free boundary condition. Using these assumed mode shape functions, mode shapes of a free-floating deep shell are illustrated. Finally, via numerical simulation, control effectiveness of distributed actuator patches with respect to various natural modes, actuator locations and other factors which influence precision control and active actuation behavior of flexible deep double curvature shell structronic systems is evaluated.
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Reports on the topic "Curvature functionals"

1

Diewert, W. Erwin, and T. J. Wales. Flexible Functional Forms and Global Curvature Conditions. Cambridge, MA: National Bureau of Economic Research, May 1989. http://dx.doi.org/10.3386/t0040.

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