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Register a new userExtrinsic Isoperimetric Analysis on Submanifolds with Curvatures Bounded from Below
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We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motion in the extrinsic balls. In those cases, where we may extend our analysis to hold all the way to infinity, we apply a capacity comparison technique to obtain a sufficient condition for the submanifolds to be parabolic, i.e. a condition which will guarantee that any Brownian particle, which is free to move around in the whole submanifold, is bound to eventually revisit any given neighborhood of its starting point with probability 1. The results of this paper are in a rough sense dual to similar results obtained previously by the present authors in complementary settings where we assume that the curvatures are bounded from above.
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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 study, from the extrinsic point of view, the structure at infinity of open submanifolds isometrically immersed in the real space forms of constant sectional curvature $kappa leq 0$. We shall use the decay of the second fundamental form of the the so-called tamed immersions to obtain a description at infinity of the submanifold in the line of the structural results in the papers Internat. Math. Res. Notices 1994, no. 9, authored by R. E. Greene, P. Petersen and S. Zhou and Math. Ann. 2001, 321 (4), authored by A. Petrunin and W. Tuschmann. We shall obtain too an estimation from below of the number of its ends in terms of the volume growth of a special class of extrinsic domains, the extrinsic balls.
Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so that under these new metrics, $K$ is the scalar curvature and $c$ is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on $K$,$c$ and $(M,g)$ we show that all the solutions of the equation can only blow up at finite points over each compact subset of $bar M$, some of them may appear on $partial M$. We describe the asymptotic behavior of the blowup solutions around each blowup point and derive an energy estimate as a consequence.
Given a Hermitian line bundle $Lto M$ over a closed, oriented Riemannian manifold $M$, we study the asymptotic behavior, as $epsilonto 0$, of couples $(u_epsilon, abla_epsilon)$ critical for the rescalings begin{align*} &E_epsilon(u, abla)=int_MBig(| abla u|^2+epsilon^2|F_ abla|^2+frac{1}{4epsilon^2}(1-|u|^2)^2Big) end{align*} of the self-dual Yang-Mills-Higgs energy, where $u$ is a section of $L$ and $ abla$ is a Hermitian connection on $L$ with curvature $F_{ abla}$. Under the natural assumption $limsup_{epsilonto 0}E_epsilon(u_epsilon, abla_epsilon)<infty$, we show that the energy measures converge subsequentially to (the weight measure $mu$ of) a stationary integral $(n-2)$-varifold. Also, we show that the $(n-2)$-currents dual to the curvature forms converge subsequentially to $2piGamma$, for an integral $(n-2)$-cycle $Gamma$ with $|Gamma|lemu$. Finally, we provide a variational construction of nontrivial critical points $(u_epsilon, abla_epsilon)$ on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgrens existence result of (nontrivial) stationary integral $(n-2)$-varifolds in an arbitrary closed Riemannian manifold.
We establish some important inequalities under a lower weighted Ricci curvature bound on Finsler manifolds. Firstly, we establish a relative volume comparison of Bishop-Gromov type. As one of the applications, we obtain an upper bound for volumes of the Finsler manifolds. Further, when the S-curvature is bounded on the whole manifold, we obtain a theorem of Bonnet-Myers type on Finsler manifolds. Finally, we obtain a sharp Poincar{e}-Lichnerowicz inequality by using integrated Bochner inequality, from which we obtain a sharp lower bound for the first eigenvalue on the Finsler manifolds.
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