This paper has been withdrawn by the authors. The theorem that we seemed to prove is false. There are stochastically incomplete properly immersed in $\mathbb{R}^{n}$
Using a deep criteria due to Pigola, Rigoli and Setti, we prove that a geodesically complete, properly immersed submanifold M of a stochastically complete Riemannian manifold N is stochastically complete. This implies that the weak Omori-Yau maximum principle holds on M. As geometric application, we prove sectional curvature estimates for properly immersed cilindrically bounded submanifolds.
Stochastic Properties of the Laplacian on Riemannian Submersions
基于Pigolla和Setti \cite{PS}的思想，我们证明了Cartan-Hadamard流形中具有有界中曲率的浸入子流形是Feller的.我们还考虑黎曼浸没$π\colon M \to N$与紧凑的最小纤维，并根据各种标准的抛物性和随机完整性，见\cite{Grygor'yan}，我们证明了$M$是费勒，抛物或stocheros完全当且仅当基地$N$是费勒，抛物或stocheros完全分别。
New rigidity results for complete non-compact spacelike submanifolds of arbitrary codimension in plane fronted waves are obtained. Under appropriate assumptions, we prove that a complete spacelike submanifold in these spacetimes is contained in a characteristic lightlike hypersurface. Moreover, for a complete codimension two extremal submanifold in a plane fronted wave we show sufficient conditions to guarantee that it is a (totally geodesic) wavefront.
We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ 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 use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong consequences in hyperbolic and Euclidean spaces. Among the consequences of our main theorem are results on submanifolds with sufficiently high index of relative nullity, submanifolds with nonpositive…
In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if the ambient manifold is of bounded geometry, we prove that such a submanifold must have at least linear volume growth. In particular, we show that a properly immersed complete noncompact hypersurface in the Euclidean space with bounded Gaussian weighted mean curvature must have …
Concerning the problem of classifying complete submanifolds of Euclidean space with codimension two admitting genuine isometric deformations, until now the only known examples with the maximal possible rank four are the real Kaehler minimal submanifolds classified by Dajczer-Gromoll \cite{dg3} in parametric form. These submanifolds behave like minimal surfaces, namely, if simple connected either they admit a nontrivial one-parameter associated family of isometric deformations or are holomorphic. In this paper, we characterize a new class of complete minimal genuinely deformable Euclidean subma…
Volume Growth, Number of Ends and the Topology of a Complete Submanifold
给定一个完全等距浸入$ϕ: P^m \longrightarrow N^n$在周围黎曼流形$N^n$中，该流形具有一个极点和径向截面曲率，该径向截面曲率由径向对称空间$M^n_w$的相应径向截面曲率从上到下界定，我们确定了关于$P$的外曲率的一组条件，该条件保证浸入是正确的，并且$P$具有有限拓扑，在论文“On Submanifold With Tamed第二基本型“（Glasgow Mathematical Journal，51，2009）中，G. Pacelli Bessa和M. Silencia Costa.当周围流形是一个径向对称空间时，证明了一个完备的极小子流形的（外在的）体积增长与它的端点数之间的不等式，这个不等式推广了Anderson的论文“The compactification of a minimal submanifold by the Gauss Map”（Preprint IEHS，1984）中关于$\erre^n$中完备的极小子流形的经典不等式.作为推论，我们得到了双曲空间中完备和极小子流形的（外在）体积增长与端点数之间的相应不等式，以及欧氏空间和双曲空间中这类子流形的Bernstein型结果，这是A. Kasue和K.徐文，“欧氏空间和双曲空间形式中某些子流形的间隙定理”，（大坂数学杂志，24，1987）。
In this note, we investigate conformally flat submanifolds of Euclidean space with positive index of relative nullity. Let $M^n$ be a complete conformally flat manifold and let $f\colon M^n\to \R^m$ be an isometric immersion. We prove the following results: (1) If the index of relative nullity is at least two, then $M^n$ is flat and $f$ is a cylinder over a flat submanifold. (2) If the scalar curvature of $M^n$ is non-negative and the index of relative nullity is positive, then $f$ is a cylinder over a submanifold with constant non-negative sectional curvature. (3) If the scalar curvature of $…
We survey the history as well as recent progress in the Yang problem concerning the existence of complete bounded complex submanifolds of the complex Euclidean spaces. We also point out some open questions on the topic.
Minimal Submanifolds of The Complex and Quaternionic Projective and Hyperbolic Spaces $\cn P^{2n-1}$, $\hn P^{2n-1}$, $\cn H^{2n-1}$, $\hn H^{2n-1}$ via Harmonic Morphisms
Let $n\ge 2$ and $k\ge 1$ be two integers. Let $M$ be an isometrically immersed closed $n$-submanifold of co-dimension $k$ that is homotopic to a point in a complete manifold $N$, where the sectional curvature of $N$ is no more than $δ<0$. We prove that the total squared mean curvature of $M$ in $N$ and the first non-zero eigenvalue $λ_1(M)$ of $M$ satisfies $$λ_1(M)\le n\left(δ+\frac{1}{\operatorname{Vol} M}\int_M |H|^2 \operatorname{dvol}\right).$$ The equality implies that $M$ is minimally immersed in a metric sphere after lifted to the universal cover of $N$. This completely settles an ope…
In this paper we establish conditions on the length of the second fundamental form of a complete minimal submanifold $M^n$ in the hyperbolic space $\mathbb{H}^{n+m}$ in order to show that $M^n$ is totally geodesic. We also obtain sharp upper bounds estimates for the first eigenvalue of the super stability operator in the case of $M$ is a surface in $\mathbb{H}^{4}$.
A relevant property of equifocal submanifolds is that their parallel sets are still immersed submanifolds, which makes them a natural generalization of the so-called isoparametric submanifolds. In this paper, we prove that the regular fibers of an analytic map $π:M^{m+k}\to B^{k}$ are equifocal whenever $M^{m+k}$ is endowed with a complete Finsler metric and there is a restriction of $π$ which is a Finsler submersion for a certain Finsler metric on the image. In addition, we prove that when the fibers provide a singular foliation on $M^{m+k}$, then this foliation is Finsler.
