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Einstein gravity, Lagrange-Finsler geometry, and nonsymmetric metrics.

Vacaru, Sergiu I. (2008)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella (2005)

Journal of the European Mathematical Society

We consider the Yamabe type family of problems $(P_{ε}) : - Δ u_{ε} = u_{ε}^{(n + 2) / (n - 2)}$ , $u_{ε} > 0$ in $A_{ε}$ , $u_{ε} = 0$ on $\partial A_{ε}$ , where $A_{ε}$ is an annulus-shaped domain of $ℝ^{n}$ , $n \geq 3$ , which becomes thinner as $ε \to 0$ . We show that for every solution $u_{ε}$ , the energy $\int_{A_{ε}} | \nabla u_{|}^{2}$ as well as the Morse index tend to infinity as $ε \to 0$ . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on $ℝ^{n}$ , a half-space or an infinite strip. Our argument also involves a Liouville type theorem...

Energy quantization for Yamabe's problem in conformal dimension.

Mahmoudi, Fethi (2006)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Entropy of eigenfunctions of the Laplacian in dimension 2

Gabriel Rivière (2010)

Journées Équations aux dérivées partielles

We study asymptotic properties of eigenfunctions of the Laplacian on compact Riemannian surfaces of Anosov type (for instance negatively curved surfaces). More precisely, we give an answer to a question of Anantharaman and Nonnenmacher [4] by proving that the Kolmogorov-Sinai entropy of a semiclassical measure $μ$ for the geodesic flow $g^{t}$ is bounded from below by half of the Ruelle upper bound. (This text has been written for the proceedings of the $37^{èmes}$ Journées EDP (Port d’Albret-June, 7-11 2010))

Estimates of the first eigenvalue of a big cup domain of a 2-sphere

Tadayuki Matsuzawa, Shukichi Tanno (1982)

Compositio Mathematica

Estimation of the visibility angle of a curve in a metric space of nonpositive curvature.

Reshetnyak, Yu.G. (2002)

Sibirskij Matematicheskij Zhurnal

Estimation of vibration frequencies of linear elastic membranes

Luca Sabatini (2018)

Applications of Mathematics

The free motion of a thin elastic linear membrane is described, in a simplyfied model, by a second order linear homogeneous hyperbolic system of partial differential equations whose spatial part is the Laplace Beltrami operator acting on a Riemannian 2-dimensional manifold with boundary. We adapt the estimates of the spectrum of the Laplacian obtained in the last years by several authors for compact closed Riemannian manifolds. To make so, we use the standard technique of the doubled manifold to...

Evolution of curvature tensors under mean curvature flow.

Tapia, Victor (2009)

Revista Colombiana de Matemáticas

Evolution of hypersurfaces by their curvature in Riemannian manifolds.

Huisken, Gerhard (1998)

Documenta Mathematica

Existence of metrics with harmonic curvature and non parallel Ricci tensor.

Chouikha, A.Raouf (2003)

Balkan Journal of Geometry and its Applications (BJGA)

Existence of positive harmonic functions on groups and on covering manifolds

Philippe Bougerol, Laure Elie (1995)

Annales de l'I.H.P. Probabilités et statistiques

Existence results for the prescribed Scalar curvature on $S^{3}$

Randa Ben Mahmoud, Hichem Chtioui (2011)

Annales de l’institut Fourier

This paper is devoted to the existence of conformal metrics on $S^{3}$ with prescribed scalar curvature. We extend well known existence criteria due to Bahri-Coron.

Extremal domains for the first eigenvalue of the Laplace-Beltrami operator

Frank Pacard, Pieralberto Sicbaldi (2009)

Annales de l’institut Fourier

We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of Laplace-Beltrami operator in some Riemannian manifold. These domains are close to geodesic spheres of small radius centered at a nondegenerate critical point of the scalar curvature.

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