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Register a new userA note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs
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In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary improving similar results from a previous work. In particular, we prove that having a pole is a necessary condition and, therefore, it can be removed as hypothesis.
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In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below and with a mild assumption at infinity, that is Gromov-Hausdorff asymptoticity to simply connected models of constant sectional curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by at most countably many isoperimetric regions sitting in some Gromov-Hausdorff limits at infinity. The Gromov-Hausdorff asymptotic analysis conducted allows us to provide, in low dimensions, a result of nonexistence of isoperimetric regions in Cartan-Hadamard manifolds that are Gromov-Hausdorff asymptotic to the Euclidean space. While studying the isoperimetric problem in the smooth setting, the nonsmooth geometry naturally emerges, and thus our treatment combines techniques from both the theories.
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 $Sigma$.
This note resolves an open problem asked by Bezrukov in the open problem session of IWOCA 2014. It shows an equivalence between regular graphs and graphs for which a sequence of invariants presents some symmetric property. We extend this result to a few other sequences.
We reinterpret the renormalized volume as the asymptotic difference of the isoperimetric profiles for convex co-compact hyperbolic 3-manifolds. By similar techniques we also prove a sharp Minkowski inequality for horospherically convex sets in $mathbb{H}^3$. Finally, we include the classification of stable constant mean curvature surfaces in regions bounded by two geodesic planes in $mathbb{H}^3$ or in cyclic quotients of $mathbb{H}^3$.
In this paper, we obtain some sufficient conditions for a 3-dimensional compact trans-Sasakian manifold of type $(alpha ,beta)$ to be homothetic to a Sasakian manifold. A characterization of a 3-dimensional cosymplectic manifold is also obtained.
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