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Journal articles on the topic 'Isolated submanifold'

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Author: Grafiati

Published: 10 March 2023

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1

Nie, Zhaohu. "The Secondary Chern–Euler Class for a General Submanifold." Canadian Mathematical Bulletin 55, no. 2 (June 1, 2012): 368–77. http://dx.doi.org/10.4153/cmb-2011-077-8.

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AbstractWe define and study the secondary Chern–Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with non-isolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern.

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2

Little, Robert D. "Regular cyclic actions on complex projective space with codimension-two fixed points." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 65, no. 1 (August 1998): 51–67. http://dx.doi.org/10.1017/s1446788700039392.

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AbstractIfM2nis a cohomologyCPnandPis an odd prime, letGpbe the cyclic group of orderp. A TypeI I0Gpaction onM2nis an action with fixed point set a codimension-2 submanifold and an isolated point. A TypeI I0Gpaction is standard if it is regular and the degree of the fixed codimension-2 submanifold is one. If n is odd and M2nadmits a standardGpaction of TypeI I0, then every TypeI I0GpactionM2nis standard and so, if n is odd,CPnadmits aGpaction of TypeI I0if and only if the action is standard.

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3

SLAPAR, MARKO. "CANCELLING COMPLEX POINTS IN CODIMENSION TWO." Bulletin of the Australian Mathematical Society 88, no. 1 (August 9, 2012): 64–69. http://dx.doi.org/10.1017/s0004972712000652.

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AbstractA generically embedded real submanifold of codimension two in a complex manifold has isolated complex points that can be classified as either elliptic or hyperbolic. In this paper we show that a pair consisting of one elliptic and one hyperbolic complex point of the same sign can be cancelled by a $\mathcal {C}^{0}$small isotopy of embeddings.

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4

Blázquez, C. Miguel. "Bifurcation from a homoclinic orbit in parabolic differential equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 103, no. 3-4 (1986): 265–74. http://dx.doi.org/10.1017/s0308210500018916.

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SynopsisThis paper considers autonomous parabolic equations which have a homoclinic orbit to an isolated equilibrium point. We study these systems under autonomous perturbations. Firstly we prove that the perturbation under which the homoclinicorbit persists forms a submanifold of codimension one. Then, if a perturbation of this manifold is considered, we prove that a unique stable periodic orbit arises from the homoclinic orbit under certain conditions for the eigenvalues of thesaddle point.

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5

Estudillo, Francisco J. M., and Alfonso Romero. "Generalized maximal surfaces in Lorentz–Minkowski space L3." Mathematical Proceedings of the Cambridge Philosophical Society 111, no. 3 (May 1992): 515–24. http://dx.doi.org/10.1017/s0305004100075587.

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In this paper we carry out a systematic study of generalized maximal surfaces in Lorentz–Minkowski space L3, with emphasis on their branch points. Roughly speaking, such a surface is given by a conformal mapping from a Riemann surface S in L3. In the last years, several authors [1, 2, 5, 6] have dealt with regular maximal surfaces in L3, i.e. with isometric immersions, with zero mean curvature, of Riemannian 2-manifolds M in L3. So, the term ‘regular’ means free of branch points. As in the minimal case, a conformal structure is naturally induced on M, which becomes a Riemann surface S. The corresponding isometric immersion is then conformal on S, and it does not have any singular points on S (i.e. points on which the differential of the mapping is not one-to-one). This is the way in which generalized maximal surfaces include regular ones. Moreover, branch points are the singular points of the conformal mapping on S. Whereas branch points of generalized minimal surfaces are isolated, we shall show in Section 2 that, in addition to isolated branch points, a generalized maximal surface in L3. may have non-isolated ones, in fact they constitute a 1-dimensional submanifold in a certain open subset of S (see Section 2). So our purpose is two-fold, firstly to study and explain in detail the branch points, and secondly to state several geometric results involving prescribed behaviour of those points on the surface.

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6

Fukui, Toshizumi, and Juan J. Nuño-Ballesteros. "Isolated roundings and flattenings of submanifolds in Euclidean spaces." Tohoku Mathematical Journal 57, no. 4 (December 2005): 469–503. http://dx.doi.org/10.2748/tmj/1140727069.

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7

Joyce, Dominic. "Special Lagrangian Submanifolds with Isolated Conical Singularities. I. Regularity." Annals of Global Analysis and Geometry 25, no. 3 (May 2004): 201–51. http://dx.doi.org/10.1023/b:agag.0000023229.72953.57.

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8

Ponsioen, S., T. Pedergnana, and G. Haller. "Analytic prediction of isolated forced response curves from spectral submanifolds." Nonlinear Dynamics 98, no. 4 (June 1, 2019): 2755–73. http://dx.doi.org/10.1007/s11071-019-05023-4.

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9

Joyce, Dominic. "Special Lagrangian Submanifolds with Isolated Conical Singularities. II. Moduli spaces." Annals of Global Analysis and Geometry 25, no. 4 (June 2004): 301–52. http://dx.doi.org/10.1023/b:agag.0000023230.21785.8d.

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10

Joyce, Dominic. "Special Lagrangian Submanifolds with Isolated Conical Singularities. v. Survey and Applications." Journal of Differential Geometry 63, no. 2 (January 2003): 279–347. http://dx.doi.org/10.4310/jdg/1090426679.

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11

Joyce, Dominic. "Special Lagrangian Submanifolds with Isolated Conical Singularities. III. Desingularization, The Unobstructed Case." Annals of Global Analysis and Geometry 26, no. 1 (August 2004): 1–58. http://dx.doi.org/10.1023/b:agag.0000023231.31950.cc.

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12

Joyce, Dominic. "Special Lagrangian Submanifolds with Isolated Conical Singularities. IV. Desingularization, Obstructions and Families." Annals of Global Analysis and Geometry 26, no. 2 (September 2004): 117–74. http://dx.doi.org/10.1023/b:agag.0000031067.19776.15.

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13

Gerner, Wadim. "Zero Set Structure of Real Analytic Beltrami Fields." Journal of Geometric Analysis 31, no. 10 (March 16, 2021): 9928–50. http://dx.doi.org/10.1007/s12220-021-00633-0.

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AbstractIn this paper, we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected 3-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected 1-dimensional submanifolds with (possibly empty) boundary and tame knots. Further, we consider the question of how complicated these tame knots can possibly be. To this end, we prove that on the standard (open) solid toroidal annulus in $${\mathbb {R}}^3$$ R 3 , there exist for any pair (p, q) of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric, there exists a real analytic Beltrami field, corresponding to the eigenvalue $$+1$$ + 1 of the curl operator, whose zero set is precisely given by a standard (p, q)-torus knot. The metrics and the corresponding Beltrami fields are constructed explicitly and can be written down in Cartesian coordinates by means of elementary functions alone.

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14

Citti, Giovanna, Gianmarco Giovannardi, and Manuel Ritoré. "Variational formulas for submanifolds of fixed degree." Calculus of Variations and Partial Differential Equations 60, no. 6 (September 22, 2021). http://dx.doi.org/10.1007/s00526-021-02100-8.

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AbstractWe consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are admissible. It turns out that the associated variational vector fields must satisfy a system of partial differential equations of first order on the submanifold. Moreover, given a vector field solution of this system, we provide a sufficient condition that guarantees the possibility of deforming the original submanifold by variations preserving its degree. As in the case of singular curves in sub-Riemannian geometry, there are examples of isolated surfaces that cannot be deformed in any direction. When the deformability condition holds we compute the Euler–Lagrange equations. The resulting mean curvature operator can be of third order.

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15

Ooi, Yuan Shyong. "Higher Codimension Minimal Submanifold with Isolated Singularity." Journal of Geometric Analysis 32, no. 5 (March 17, 2022). http://dx.doi.org/10.1007/s12220-022-00904-4.

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16

Jang, Donghoon. "Almost Complex Torus Manifolds—Graphs and Hirzebruch Genera." International Mathematics Research Notices, August 25, 2022. http://dx.doi.org/10.1093/imrn/rnac237.

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Abstract Let a $k$-dimensional torus $T^k$ act on a $2n$-dimensional compact connected almost complex manifold $M$ with isolated fixed points. As in the case of circle actions, we show that there exists a (directed labeled) multigraph that contains information on weights at the fixed points and isotropy submanifolds of $M$. This includes the notion of a GKM (Goresky-Kottwitz-MacPherson) graph as a special case that weights at each fixed point are pairwise linearly independent. If in addition $k=n$, that is, $M$ is an almost complex torus manifold, the multigraph is a graph; it has no multiple edges. We show that the Hirzebruch $\chi _y$-genus $\chi _y(M)=\sum _{i=0}^n a_i(M) \cdot (-y)^i$ of an almost complex torus manifold $M$ satisfies $a_i(M)> 0$ for $0 \leq i \leq n$. In particular, the Todd genus of $M$ is positive and there are at least $n+1$ fixed points.

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17

Schorlepp, Timo, Tobias Grafke, and Rainer Grauer. "Symmetries and Zero Modes in Sample Path Large Deviations." Journal of Statistical Physics 190, no. 3 (January 9, 2023). http://dx.doi.org/10.1007/s10955-022-03051-w.

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AbstractSharp large deviation estimates for stochastic differential equations with small noise, based on minimizing the Freidlin–Wentzell action functional under appropriate boundary conditions, can be obtained by integrating certain matrix Riccati differential equations along the large deviation minimizers or instantons, either forward or backward in time. Previous works in this direction often rely on the existence of isolated minimizers with positive definite second variation. By adopting techniques from field theory and explicitly evaluating the large deviation prefactors as functional determinant ratios using Forman’s theorem, we extend the approach to general systems where degenerate submanifolds of minimizers exist. The key technique for this is a boundary-type regularization of the second variation operator. This extension is particularly relevant if the system possesses continuous symmetries that are broken by the instantons. We find that removing the vanishing eigenvalues associated with the zero modes is possible within the Riccati formulation and amounts to modifying the initial or final conditions and evaluation of the Riccati matrices. We apply our results in multiple examples including a dynamical phase transition for the average surface height in short-time large deviations of the one-dimensional Kardar–Parisi–Zhang equation with flat initial profile.

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