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Register a new userMean time exit and isoperimetric inequalities for minimal submanifolds of $Ntimes mathbb{R}$
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Greg\\'orio Pacelli F. Bessa
Publication date
2008
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English
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Based on Markvorsen and Palmers work on mean time exit and isoperimetric inequalities we establish slightly better isoperimetric inequalities and mean time exit estimates for minimal submanifolds of $Ntimesmathbb{R}$. We also prove isoperimetric inequalities for submanifolds of Hadamard spaces with tamed second fundamental form.
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Let $Sigma$ be a $k$-dimensional complete proper minimal submanifold in the Poincar{e} ball model $B^n$ of hyperbolic geometry. If we consider $Sigma$ as a subset of the unit ball $B^n$ in Euclidean space, we can measure the Euclidean volumes of the given minimal submanifold $Sigma$ and the ideal boundary $partial_infty Sigma$, say $rvol(Sigma)$ and $rvol(partial_infty Sigma)$, respectively. Using this concept, we prove an optimal linear isoperimetric inequality. We also prove that if $rvol(partial_infty Sigma) geq rvol(mathbb{S}^{k-1})$, then $Sigma$ satisfies the classical isoperimetric inequality. By proving the monotonicity theorem for such $Sigma$, we further obtain a sharp lower bound for the Euclidean volume $rvol(Sigma)$, which is an extension of Fraser and Schoens recent result cite{FS} to hyperbolic space. Moreover we introduce the M{o}bius volume of $Sigma$ in $B^n$ to prove an isoperimetric inequality via the M{o}bius volume for $Sigma$.
We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $ N times mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N} leq kappa$. When the immersion is minimal our estimates are sharp. We also show that cylindrically bounded minimal surfaces has positive fundamental tone.
By studying the monotonicity of the first nonzero eigenvalues of Laplace and p-Laplace operators on a closed convex hypersurface $M^n$ which evolves under inverse mean curvature flow in $mathbb{R}^{n+1}$, the isoperimetric lower bounds for both eigenvalues were founded.
In this paper we provide an extension to the Jellett-Minkowskis formula for immersed submanifolds into ambient manifolds which possesses a pole and radial curvatures bounded from above or below by the radial sectional curvatures of a rotationally symmetric model space. Using this Jellett-Minkowskis generalized formula we can focus on several isoperimetric problems. More precisely, on lower bounds for isoperimetric quotients of any precompact domain with smooth boundary, or on the isoperimetric profile of the submanifold and its modified volume. In the particular case of a model space with strictly decreasing radial curvatures, an Aleksandrov type theorem is provided.
We establish curvature estimates and a convexity result for mean convex properly embedded $[varphi,vec{e}_{3}]$-minimal surfaces in $mathbb{R}^3$, i.e., $varphi$-minimal surfaces when $varphi$ depends only on the third coordinate of $mathbb{R}^3$. Led by the works on curvature estimates for surfaces in 3-manifolds, due to White for minimal surfaces, to Rosenberg, Souam and Toubiana, for stable CMC surfaces, and to Spruck and Xiao for stable translating solitons in $mathbb{R}^3$, we use a compactness argument to provide curvature estimates for a family of mean convex $[varphi,vec{e}_{3}]$-minimal surfaces in $mathbb{R}^{3}$. We apply this result to generalize the convexity property of Spruck and Xiao for translating solitons. More precisely, we characterize the convexity of a properly embedded $[varphi,vec{e}_{3}]$-minimal surface in $mathbb{R}^{3}$ with non positive mean curvature when the growth at infinity of $varphi$ is at most quadratic.
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