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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.
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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.
We investigate complete minimal submanifolds $fcolon M^3toHy^n$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in cite{dksv1} and cite{dksv2}, respectively. If the scalar curvature is bounded from below we conclude that the submanifold has to be either totally geodesic or a generalized cone over a complete minimal surface lying in an equidistant submanifold of $Hy^n$.
We give a set of sufficient and necessary conditions for parabolicity and hyperbolicity of a submanifold with controlled mean curvature in a Riemannian manifold with a pole and with sectional curvatures bounded from above or from below.
We use drifted Brownian motion in warped product model spaces as comparison constructions to show $p$-hyperbolicity of a large class of submanifolds for $pge 2$. The condition for $p$-hyperbolicity is expressed in terms of upper support functions for the radial sectional curvatures of the ambient space and for the radial convexity of the submanifold. In the process of showing $p$-hyperbolicity we also obtain explicit lower bounds on the $p$-capacity of finite annular domains of the submanifolds in terms of the drifted 2-capacity of the corresponding annuli in the respective comparison spaces.
In this paper we investigate $m$-dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least $m-2$ at any point. These are austere submanifolds in the sense of Harvey and Lawson cite{harvey} and were initially studied by Bryant cite{br}. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit cite{DF2} in terms of a class of surfaces, called elliptic, for which the ellipse of curvature of a certain order is a circle at any point. Under the assumption of completeness, it turns out that any submanifold is either totally geodesic or has dimension three. In the latter case there are plenty of examples, even compact ones. Under the mild assumption that the Omori-Yau maximum principle holds on the manifold, a trivial condition in the compact case, we provide a complete local parametric description of the submanifolds in terms of $1$-isotropic surfaces in Euclidean space. These are the minimal surfaces for which the standard ellipse of curvature is a circle at any point. For these surfaces, there exists a Weierstrass type representation that generates all simply-connected ones.
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