THE (p,m)-WIDTH OF RIEMANNIAN MANIFOLDS AND ITS REALIZATION
نویسندگان
چکیده
While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Marques and Neves used the p-width to prove the existence of infinite many minimal hypersurfaces in compact manifolds with positive Ricci curvature. However, whether the p-width can be realized as the volume of minimal hypersurfaces is not known yet. We introduced the concept of the (p,m)width which can be viewed as the stratification of the p-width, and proved that the (p,m)-width can be realized as the volume of minimal hypersurfaces with multiplicities.
منابع مشابه
Evolution of the first eigenvalue of buckling problem on Riemannian manifold under Ricci flow
Among the eigenvalue problems of the Laplacian, the biharmonic operator eigenvalue problems are interesting projects because these problems root in physics and geometric analysis. The buckling problem is one of the most important problems in physics, and many studies have been done by the researchers about the solution and the estimate of its eigenvalue. In this paper, first, we obtain the evol...
متن کاملGEOMETRIZATION OF HEAT FLOW ON VOLUMETRICALLY ISOTHERMAL MANIFOLDS VIA THE RICCI FLOW
The present article serves the purpose of pursuing Geometrization of heat flow on volumetrically isothermal manifold by means of RF approach. In this article, we have analyzed the evolution of heat equation in a 3-dimensional smooth isothermal manifold bearing characteristics of Riemannian manifold and fundamental properties of thermodynamic systems. By making use of the notions of various curva...
متن کاملOn Rigidity of Grauert Tubes
Given a real-analytic Riemannian manifold M there exists a canonical complex structure on part of its tangent bundle which turns leaves of the Riemannian foliation on TM into holomorphic curves. A Grauert tube over M of radius r, denoted as T rM , is the collection of tangent vectors of M of length less than r equipped with this canonical complex structure. We say the Grauert tube T rM is rigid...
متن کاملLipshitz Maps from Surfaces
We give a simple procedure to estimate the smallest Lipshitz constant of a degree 1 map from a Riemannian 2-sphere to the unit 2-sphere, up to a factor of 10. Using this procedure, we are able to prove several inequalities involving this Lipshitz constant. For instance, if the smallest Lipshitz constant is at least 1, then the Riemannian 2-sphere has Uryson 1-Width less than 12 and contains a c...
متن کاملMetric transformations under collapsing of Riemannian manifolds
Gromov-Hausdorff convergence is an important tool in comparison Riemannian geometry. Given a sequence of Riemannian manifolds of dimension n with Ricci curvature bounded from below, Gromov’s precompactness theorem says that a subsequence will converge in the pointed Gromov-Hausdorff topology to a length space [G-99, Section 5A]. If the sequence has bounded sectional curvature, then the limit wi...
متن کامل