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Journal articles on the topic 'Curvature bounds'

Author: Grafiati

Published: 10 March 2023

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1

Bessa, Gregório P., Luquésio P. Jorge, Barnabé P. Lima, and José F. Montenegro. "Fundamental tone estimates for elliptic operators in divergence form and geometric applications." Anais da Academia Brasileira de Ciências 78, no. 3 (September 2006): 391–404. http://dx.doi.org/10.1590/s0001-37652006000300001.

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We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+ 1(c) with Hr + 1 > 0 in terms of the r-th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero Lr-eigenvalue of a closed hypersurface of Nn+ 1(c).

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2

Peters, Jörg, and Georg Umlauf. "Computing curvature bounds for bounded curvature subdivision." Computer Aided Geometric Design 18, no. 5 (June 2001): 455–61. http://dx.doi.org/10.1016/s0167-8396(01)00041-3.

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3

Sabatini, Luca. "Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary II." Analele Universitatii "Ovidius" Constanta - Seria Matematica 28, no. 1 (March 1, 2020): 165–79. http://dx.doi.org/10.2478/auom-2020-0012.

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AbstractWe present some estimate of the Laplacian Spectrum and of Topological Invariants for Riemannian manifold with pinched sectional curvature and with non-empty and non-convex boundary with finite injectivity radius. These estimates do not depend directly on the the lower bound of the boundary injectivity radius but on the bounds of the curvatures of the manifold and its boundary.

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4

Frenck, Georg, and Jan-Bernhard Kordaß. "Spaces of positive intermediate curvature metrics." Geometriae Dedicata 214, no. 1 (June 23, 2021): 767–800. http://dx.doi.org/10.1007/s10711-021-00635-w.

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AbstractIn this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional $$\mathrm {Spin}$$ Spin -manifold have many non-trivial homotopy groups provided that the manifold admits such a metric.

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5

Erbar, Matthias, and Martin Huesmann. "Curvature bounds for configuration spaces." Calculus of Variations and Partial Differential Equations 54, no. 1 (November 19, 2014): 397–430. http://dx.doi.org/10.1007/s00526-014-0790-1.

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6

Lytchak, Alexander, and Stephan Stadler. "Improvements of upper curvature bounds." Transactions of the American Mathematical Society 373, no. 10 (August 5, 2020): 7153–66. http://dx.doi.org/10.1090/tran/8123.

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7

Kapovitch, Vitali. "Curvature bounds via Ricci smoothing." Illinois Journal of Mathematics 49, no. 1 (January 2005): 259–63. http://dx.doi.org/10.1215/ijm/1258138317.

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8

Hu, Zisheng, and Senlin Xu. "Bounds on the fundamental groups with integral curvature bound." Geometriae Dedicata 134, no. 1 (April 19, 2008): 1–16. http://dx.doi.org/10.1007/s10711-008-9235-3.

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9

Lott, John. "On scalar curvature lower bounds and scalar curvature measure." Advances in Mathematics 408 (October 2022): 108612. http://dx.doi.org/10.1016/j.aim.2022.108612.

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10

Wang, Xu-Jia, John Urbas, and Weimin Sheng. "Interior curvature bounds for a class of curvature equations." Duke Mathematical Journal 123, no. 2 (June 2004): 235–64. http://dx.doi.org/10.1215/s0012-7094-04-12321-8.

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11

Alkhaldi, Ali, Mohd Aquib, Aliya Siddiqui, and Mohammad Shahid. "Pinching Theorems for Statistical Submanifolds in Sasaki-Like Statistical Space Forms." Entropy 20, no. 9 (September 11, 2018): 690. http://dx.doi.org/10.3390/e20090690.

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In this paper, we obtain the upper bounds for the normalized δ -Casorati curvatures and generalized normalized δ -Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds with constant curvature. Further, we discuss the equality case of the inequalities. Moreover, we give the necessary and sufficient condition for a Sasaki-like statistical manifold to be η -Einstein. Finally, we provide the condition under which the metric of Sasaki-like statistical manifolds with constant curvature is a solution of vacuum Einstein field equations.

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12

Liang, Fei-Tsen. "Boundary Regularity for Capillary Surfaces." gmj 12, no. 2 (June 2005): 283–307. http://dx.doi.org/10.1515/gmj.2005.283.

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Abstract For solutions of capillarity problems with the boundary contact angle being bounded away from 0 and π and the mean curvature being bounded from above and below, we show the Lipschitz continuity of a solution up to the boundary locally in any neighborhood in which the solution is bounded and ∂Ω is 𝐶2; the Lipschitz norm is determined completely by the upper bound of | cos θ|, together with the lower and upper bounds of 𝐻, the upper bound of the absolute value of the principal curvatures of ∂Ω and the dimension 𝑛.

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13

Carreras, Francisco J., Fernando Giménez, and Vicente Miquel. "A comparison theorem for the first Dirichlet eigenvalue of a domain in a Kaehler submanifold." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 56, no. 2 (April 1994): 267–77. http://dx.doi.org/10.1017/s144678870003487x.

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AbstractWe give a sharp lower bound for the first eigenvalue of the Dirichlet eigenvalue problem on a domain of a complex submanifold of a Kaehler manifold with curvature bounded from above. The bound on the first eigenvalue is given as a function of the extrinsic outer radius and the bounds on the curvature, and it is attained only on geodesic spheres of a space of constant holomorphic sectional curvature embedded in the Kaehler manifold as a totally geodesic submanifold.

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14

Ambrosio, Luigi, Nicola Gigli, and Giuseppe Savaré. "Bakry–Émery curvature-dimension condition and Riemannian Ricci curvature bounds." Annals of Probability 43, no. 1 (February 2015): 339–404. http://dx.doi.org/10.1214/14-aop907.

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15

Liu, Shiping, Florentin Münch, Norbert Peyerimhoff, and Christian Rose. "Distance Bounds for Graphs with Some Negative Bakry-Émery Curvature." Analysis and Geometry in Metric Spaces 7, no. 1 (March 1, 2019): 1–14. http://dx.doi.org/10.1515/agms-2019-0001.

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Abstract We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the tubular neighborhood with an explicit radius around the non-positively curved vertices. Those results seem to be the first assuming non-constant Bakry-Émery curvature assumptions on graphs.

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16

Lytchak, Alexander, and Stefan Wenger. "Isoperimetric characterization of upper curvature bounds." Acta Mathematica 221, no. 1 (2018): 159–202. http://dx.doi.org/10.4310/acta.2018.v221.n1.a5.

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17

Ketterer, Christian. "Ricci curvature bounds for warped products." Journal of Functional Analysis 265, no. 2 (July 2013): 266–99. http://dx.doi.org/10.1016/j.jfa.2013.05.008.

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18

Borzellino, Joseph E. "Orbifolds with lower Ricci curvature bounds." Proceedings of the American Mathematical Society 125, no. 10 (1997): 3011–18. http://dx.doi.org/10.1090/s0002-9939-97-04046-x.

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19

Peterson, P., S. D. Shteingold, and G. Wei. "Comparison Geometry with Integral Curvature Bounds." Geometric And Functional Analysis 7, no. 6 (December 1, 1997): 1011–30. http://dx.doi.org/10.1007/s000390050035.

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20

Sprouse, Chadwick. "Integral curvature bounds and bounded diameter." Communications in Analysis and Geometry 8, no. 3 (2000): 531–43. http://dx.doi.org/10.4310/cag.2000.v8.n3.a4.

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21

Villani, Cédric. "Synthetic theory of Ricci curvature bounds." Japanese Journal of Mathematics 11, no. 2 (August 29, 2016): 219–63. http://dx.doi.org/10.1007/s11537-016-1531-3.

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22

Fang, Hao, and Mijia Lai. "Volume bounds of conic 2-spheres." International Journal of Mathematics 29, no. 02 (February 2018): 1850010. http://dx.doi.org/10.1142/s0129167x18500106.

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We obtain sharp volume bound for a conic 2-sphere in terms of its Gaussian curvature bound. We also give the geometric models realizing the extremal volume. In particular, when the curvature is bounded in absolute value by [Formula: see text], we compute the minimal volume of a conic sphere in the sense of Gromov. In order to apply the level set analysis and isoperimetric inequality as in our previous works, we develop new analytical tools to treat regions with vanishing curvature.

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23

Kloeckner, Benoît, and Stéphane Sabourau. "Mixed sectional-Ricci curvature obstructions on tori." Journal of Topology and Analysis 12, no. 03 (October 23, 2018): 713–34. http://dx.doi.org/10.1142/s1793525319500626.

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We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp’s theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional curvature of these metrics cannot be bounded from above by an arbitrarily small positive constant. In particular, if the Ricci curvature of a Riemannian torus is negative, bounded away from zero, then there exist some planar directions in this torus where the sectional curvature is positive, bounded away from zero. All constants are explicit and depend only on the dimension of the torus.

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24

Aquib, Mohd, and Mohammad Shahid. "Bounds for generalized normalized δ-Casorati curvatures for submanifolds in Bochner Kaehler manifold." Filomat 32, no. 2 (2018): 693–704. http://dx.doi.org/10.2298/fil1802693a.

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In this paper, we prove sharp inequalities between the normalized scalar curvature and the generalized normalized ?-Casorati curvatures for different submanifolds in Bochner Kaehler manifold. Moreover, We also characterize submanifolds on which the equalities hold.

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25

Siddiqui, Aliya Naaz, Mohd Danish Siddiqi, and Ali Hussain Alkhaldi. "Bounds for Statistical Curvatures of Submanifolds in Kenmotsu-like Statistical Manifolds." Mathematics 10, no. 2 (January 6, 2022): 176. http://dx.doi.org/10.3390/math10020176.

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In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated.

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26

BESSA, G. P., and J. F. MONTENEGRO. "On Cheng's eigenvalue comparison theorem." Mathematical Proceedings of the Cambridge Philosophical Society 144, no. 3 (May 2008): 673–82. http://dx.doi.org/10.1017/s0305004107000965.

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AbstractWe observe that Cheng's Eigenvalue Comparison Theorem for normal geodesic balls [4] is still valid if we impose bounds on the mean curvature of the distance spheres instead of bounds on the sectional and Ricci curvatures. In this version, there is a weak form of rigidity in case of equality of the eigenvalues. Namely, equality of the eigenvalues implies that the distance spheres of the same radius on each ball has the same mean curvature. On the other hand, we construct smooth metrics $g_{\kappa}$ on $[0,r]\times \mathbb{S}^{3}$, non-isometric to the standard metric canκ of constant sectional curvature κ, such that the geodesic balls $B_{g_{\kappa}}(r)=([0,r]\times \mathbb{S}^{3},g_{\kappa})$, $B_{{\rm can}_{\kappa}}(r)=([0,r]\times \mathbb{S}^{3},{\rm can}_{\kappa})$ have the same first eigenvalue, the same volume and the distance spheres $\partial B_{g_{\kappa}}(s)$ and$\partial B_{{\rm can}_{\kappa}}(s)$, $0<s\leq r$, have the same mean curvatures. In the end, we apply this version of Cheng's Eigenvalue Comparison Theorem to construct examples of Riemannian manifolds M with arbitrary topology with positive fundamental tone λ*(M)>0 extending Veeravalli's examples,[7]

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27

YUN, Jong-Gug. "A SPHERE THEOREM WITH INTEGRAL CURVATURE BOUNDS." Kyushu Journal of Mathematics 56, no. 2 (2002): 225–34. http://dx.doi.org/10.2206/kyushujm.56.225.

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28

Dessai, Anand, Wilderich Tuschmann, and Burkhard Wilking. "Mini-Workshop: Manifolds with Lower Curvature Bounds." Oberwolfach Reports 9, no. 1 (2012): 5–42. http://dx.doi.org/10.4171/owr/2012/01.

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29

Chai, Y. D., Yong-Il Kim, and Doo-Hann Lee. "CURVATURE BOUNDS OF EUCLIDEAN CONES OF SPHERES." Bulletin of the Korean Mathematical Society 40, no. 2 (May 1, 2003): 319–26. http://dx.doi.org/10.4134/bkms.2003.40.2.319.

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30

Torromé, Ricardo Gallego. "Maximal acceleration geometries and spacetime curvature bounds." International Journal of Geometric Methods in Modern Physics 17, no. 04 (March 2020): 2050060. http://dx.doi.org/10.1142/s0219887820500607.

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A geometric framework for metrics of maximal acceleration which is applicable to large proper accelerations is discussed, including a theory of connections associated with the geometry of maximal acceleration. In such a framework, it is shown that the uniform bound on the proper maximal acceleration implies a uniform bound for certain bilinear combinations of the Riemannian curvature components in the domain of the spacetime where curvature is finite.

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31

Guijarro, Luis, and Frederick Wilhelm. "Focal radius, rigidity, and lower curvature bounds." Proceedings of the London Mathematical Society 116, no. 6 (February 13, 2018): 1519–52. http://dx.doi.org/10.1112/plms.12113.

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32

Bondaryk, Joseph E., Phillip Abbot, Charles Gedney, and Edward Sullivan. "Performance bounds on wave‐front curvature ranging." Journal of the Acoustical Society of America 113, no. 4 (April 2003): 2213. http://dx.doi.org/10.1121/1.4780246.

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33

Fang, Hao, and Weifeng Wo. "On singular curves with Gaussian curvature bounds." Analysis 38, no. 3 (August 1, 2018): 127–36. http://dx.doi.org/10.1515/anly-2017-0045.

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Abstract In this short note, we consider piece-wise smooth strictly convex curves in {\mathbb{R}^{2}} with prescribed singular angles. Given some geodesic curvature bounds, we give the sharp length estimates.

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34

Cheeger, Jeff, Tobias H. Colding, and Gang Tian. "Constraints on singularities under Ricci curvature bounds." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 324, no. 6 (March 1997): 645–49. http://dx.doi.org/10.1016/s0764-4442(97)86982-0.

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35

Dai, Xianzhe, Guofang Wei, and Rugang Ye. "Smoothing Riemannian metrics with Ricci curvature bounds." Manuscripta Mathematica 90, no. 1 (December 1996): 49–61. http://dx.doi.org/10.1007/bf02568293.

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36

Woolgar, Eric, and William Wylie. "Curvature-dimension bounds for Lorentzian splitting theorems." Journal of Geometry and Physics 132 (October 2018): 131–45. http://dx.doi.org/10.1016/j.geomphys.2018.06.001.

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37

Cavalletti, Fabio, and Andrea Mondino. "Measure rigidity of Ricci curvature lower bounds." Advances in Mathematics 286 (January 2016): 430–80. http://dx.doi.org/10.1016/j.aim.2015.09.016.

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38

Petersen, P., and G. Wei. "Relative Volume Comparison with Integral Curvature Bounds." Geometric And Functional Analysis 7, no. 6 (December 1, 1997): 1031–45. http://dx.doi.org/10.1007/s000390050036.

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39

Bakry, Dominique, and Olfa Zribi. "Curvature dimension bounds on the deltoid model." Annales de la Faculté des sciences de Toulouse : Mathématiques 25, no. 1 (February 29, 2016): 65–90. http://dx.doi.org/10.5802/afst.1487.

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40

Treibergs, Andrejs. "Bounds for hyperspheres of prescribed Gaussian curvature." Journal of Differential Geometry 31, no. 3 (1990): 913–26. http://dx.doi.org/10.4310/jdg/1214444638.

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41

Petersen, Peter, and Chadwick Sprouse. "Integral curvature bounds, distance estimates and applications." Journal of Differential Geometry 50, no. 2 (1998): 269–98. http://dx.doi.org/10.4310/jdg/1214461171.

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42

Tam, Luen-Fai, and Chengjie Yu. "Complex Product Manifolds and Bounds of Curvature." Asian Journal of Mathematics 14, no. 2 (2010): 235–42. http://dx.doi.org/10.4310/ajm.2010.v14.n2.a4.

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43

Honda, Shouhei. "On the spaces with Ricci curvature bounds." Sugaku Expositions 32, no. 1 (March 21, 2019): 87–112. http://dx.doi.org/10.1090/suga/439.

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44

Anderson, Michael T. "Regularity for Lorentz metrics under curvature bounds." Journal of Mathematical Physics 44, no. 7 (2003): 2994. http://dx.doi.org/10.1063/1.1580199.

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45

Schwachhöfer, Lorenz J. "Lower curvature bounds and cohomogeneity one manifolds." Differential Geometry and its Applications 17, no. 2-3 (September 2002): 209–28. http://dx.doi.org/10.1016/s0926-2245(02)00108-0.

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46

Santos-Rodríguez, Jaime. "Invariant Measures and Lower Ricci Curvature Bounds." Potential Analysis 53, no. 3 (July 11, 2019): 871–97. http://dx.doi.org/10.1007/s11118-019-09790-y.

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47

Bonk, Mario, and Thomas Foertsch. "Asymptotic upper curvature bounds in coarse geometry." Mathematische Zeitschrift 253, no. 4 (March 28, 2006): 753–85. http://dx.doi.org/10.1007/s00209-005-0931-5.

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48

Azarhooshang, Nazanin, Prithviraj Sengupta, and Bhaskar DasGupta. "A Review of and Some Results for Ollivier–Ricci Network Curvature." Mathematics 8, no. 9 (August 24, 2020): 1416. http://dx.doi.org/10.3390/math8091416.

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Characterizing topological properties and anomalous behaviors of higher-dimensional topological spaces via notions of curvatures is by now quite common in mainstream physics and mathematics, and it is therefore natural to try to extend these notions from the non-network domains in a suitable way to the network science domain. In this article we discuss one such extension, namely Ollivier’s discretization of Ricci curvature. We first motivate, define and illustrate the Ollivier–Ricci Curvature. In the next section we provide some “not-previously-published” bounds on the exact and approximate computation of the curvature measure. In the penultimate section we review a method based on the linear sketching technique for efficient approximate computation of the Ollivier–Ricci network curvature. Finally in the last section we provide concluding remarks with pointers for further reading.

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49

Bamler, Richard H., and Qi S. Zhang. "Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature." Advances in Mathematics 319 (October 2017): 396–450. http://dx.doi.org/10.1016/j.aim.2017.08.025.

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50

Perales, Raquel. "Volumes and limits of manifolds with Ricci curvature and mean curvature bounds." Differential Geometry and its Applications 48 (October 2016): 23–37. http://dx.doi.org/10.1016/j.difgeo.2016.05.004.

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