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Topological Methods in Nonlinear Analysis

A note on local minimizers of energy on complete manifolds
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A note on local minimizers of energy on complete manifolds

Authors

  • Márcio Batista https://orcid.org/0000-0002-6495-3842
  • José I. Santos

DOI:

https://doi.org/10.12775/TMNA.2022.013

Keywords

Stability, minimizer, manifolds

Abstract

In this paper, we study the geometric rigidity of complete Riemannian manifolds admitting local minimizers of energy functionals. More precisely, assuming the existence of a non-trivial local minimizer and under suitable assumptions, a Riemannian manifold under consideration must be a product manifold furnished with a warped metric. Secondly, under similar hypotheses, we deduce a geometrical splitting in the same fashion as in the Cheeger-Gromoll splitting theorem and we also get information about local minimizers.

References

L. Ambrosio and X. Cabré, Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 (2000) 725–739.

D. Bakry and M. Émery, Diffusions hypercontractives. Séminaire de probabilities, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp.1̇77–206.

S.-Y.A. Chang, M.J. Gursky and P. Yang, Conformal invariants associated to a measure, Proc. Natl. Acad. Sci. USA 103 (2006), no. 8, 2535–2540.

E. De Giorgi, Convergence problems for functionals and operators, Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Pitagora, Bologna, 1979, pp. 131–188.

M. del Pino, M. Kowalczyk and J. Wei, On De Giorgi’s conjecture in dimension N ≥ 9, Ann. of Math. (2) 174 (2011), no. 3, 1485–1569.

A. Farina and E. Valdinoci, The state of the art for a conjecture of De Giorgi and related problems, Recent Progress on Reaction – Diffusion Systems and Viscosity Solutions, World Sci. Publ., Hackensack, NJ, 2009, pp. 74–96.

A. Farina, L. Mari and E. Valdinoci, Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds, Comm. Partial Differential Equations 38 (2013), no. 10, 1818–1862.

D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199–211.

K. Frensel, Stable complete surfaces with constant mean curvature, Bol. Soc. Brasil. Mat. (N.S.) 27 (1996), no. 2, 129–144.

G. Friedman, E. Hunsicker, A. Libgober and L. Maxim, Topology of Stratified Spaces, MSRI Publications, vol. 58, USA, 2011.

N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998) 481–491.

N. Ghoussoub and C. Gui, On De Giorgi’s conjecture in dimensions 4 and 5., Ann. of Math. (2) 157 (2003), no. 1, 313–334.

A. Grigor’yan, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, vol. 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009.

A. Lichnerowich, Variétés riemanniennes à tenseur C non négatif, C.R. Acad. Sci. Paris Sér. A–B 271 (1970), A650–A653.

A. Lichnerowich, Variétés kählériennes à première classe de Chern non negative et variétés riemanniennes à courbure de Ricci généralisée non negative, J. Differential Geom. 6 (1971/1972), 47–94.

O. Monteanu and J. Wang, Analysis of weighted Laplacian and applications to Ricci solitons, Comm. Anal. Geom. 20 (2012), no. 1, 55–94.

O. Monteanu and J. Wang, Geometry of Manifolds with Densities, Adv. Math. 259 (2014), 269–305.

F. Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), no. 8, 853–858.

F. Pacard and J. Wei, Stable solutions of the Allen–Cahn equation in dimension 8 and minimal cones, J. Funct. Anal. 264 (2013), no. 5, 1131–1167.

S. Pigola, M. Rigoli and A. G. Setti, Vanishing and Finiteness Results in Geometric Analysis. A Generalization of the Böchner technique, Progress in Math., vol. 266, Birkäuser, 2008.

G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math/0211159 [math.DG], (2002).

H. Rosenberg, Constant mean curvature surfaces in homogeneously regular 3-manifolds, Bull. Austral. Math. Soc. 74 (2006), no. 2, 227–238.

O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math. (2) 169 (2009), no. 1, 41–78.

M. Troyanov, Parabolicity of manifolds, Siberian Adv. Math. 9 (1999), no. 4, 125–150.

G. Wei and W. Wylie, Comparison geometry for the Bakry–Emery–Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377–405.

W. Wylie, Sectional Curvature for Riemannian Manifolds with Density, Geometriae Dedicata, vol 178, Issue 1, 2015, pp. 151–169.

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Published

2022-12-10

How to Cite

1.
BATISTA, Márcio & SANTOS, José I. A note on local minimizers of energy on complete manifolds. Topological Methods in Nonlinear Analysis [online]. 10 December 2022, T. 60, nr 2, s. 565–579. [accessed 25.3.2023]. DOI 10.12775/TMNA.2022.013.
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Vol 60, No 2 (December 2022)

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Articles

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Copyright (c) 2022 Márcio Batista, José I. Santos

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This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.

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