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Register a new userComplete minimal submanifolds with nullity in the hyperbolic space
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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 $Hy^n$.
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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.
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 $Sigma$.
In this paper, we investigate geometric conditions for isometric immersions with positive index of relative nullity to be cylinders. There is an abundance of noncylindrical $n$-dimensional minimal submanifolds with index of relative nullity $n-2$, fully described by Dajczer and Florit cite{DF2} in terms of a certain class of elliptic surfaces. Opposed to this, we prove that nonminimal $n$-dimensional submanifolds in space forms of any codimension are locally cylinders provided that they carry a totally geodesic distribution of rank $n-2geq2,$ which is contained in the relative nullity distribution, such that the length of the mean curvature vector field is constant along each leaf. The case of dimension $n=3$ turns out to be special. We show that there exist elliptic three-dimensional submanifolds in spheres satisfying the above properties. In fact, we provide a parametrization of three-dimensional submanifolds as unit tangent bundles of minimal surfaces in the Euclidean space whose first curvature ellipse is nowhere a circle and its second one is everywhere a circle. Moreover, we provide several applications to submanifolds whose mean curvature vector field has constant length, a much weaker condition than being parallel.
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 give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension $mgeq 3$ of a product $Q_{k_1}^{n_1}times Q_{k_2}^{n_2}$ of two space forms whose curvatures satisfy $k_1+k_2 eq 0$ to the classification of $m$-dimensional umbilical submanifolds of codimension two of $Sf^ntimes R$ and $Hy^ntimes R$. The case of $Sf^ntimes R$ was carried out in cite{mt}. As a main tool we derive reduction of codimension theorems of independent interest for submanifolds of products of two space forms.
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