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Register a new userParabolicity, Brownian escape rate and properness of self-similar solutions of the direct and inverse Mean Curvature Flow
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We study some potential theoretic properties of homothetic solitons $Sigma^n$ of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in $mathbb{R}^{n+m}$, we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons $Sigma^n$ with $n>2$ are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the Mean Exit Time function for the Brownian motion defined on $Sigma$ as well as a classification of properly immersed MCF-self-shrinkers with bounded second fundamental form, following the lines of cite{CaoLi}.
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We will give a new proof of a recent result of P.~Daskalopoulos, G.Huisken and J.R.King ([DH] and reference [7] of [DH]) on the existence of self-similar solution of the inverse mean curvature flow which is the graph of a radially symmetric solution in $mathbb{R}^n$, $nge 2$, of the form $u(x,t)=e^{lambda t}f(e^{-lambda t} x)$ for any constants $lambda>frac{1}{n-1}$ and $mu<0$ such that $f(0)=mu$. More precisely we will give a new proof of the existence of a unique radially symmetric solution $f$ of the equation $mbox{div},left(frac{ abla f}{sqrt{1+| abla f|^2}} right)=frac{1}{lambda}cdotfrac{sqrt{1+| abla f|^2}}{xcdot abla f-f}$ in $mathbb{R}^n$, $f(0)=mu$, for any $lambda>frac{1}{n-1}$ and $mu<0$, which satisfies $f_r(r)>0$, $f_{rr}(r)>0$ and $rf_r(r)>f(r)$ for all $r>0$. We will also prove that $lim_{rtoinfty}frac{rf_r(r)}{f(r)}=frac{lambda (n-1)}{lambda (n-1)-1}$.
We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m leq pleq infty$, and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its properness. Finally, we relate the curvature and the topology of a complete and non-compact $2$-Riemannian manifold $M$ with non-positive Gaussian curvature and finite topology, using the study of the focal points of the transverse Jacobi fields to a geodesic ray in $M$ . In particular, we have explored the relation between the minimal focal distance of a geodesic ray and the total curvature of an end containing that geodesic ray.
As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. Recently, Guo cite{Guo} proved that $n$-dimensional compact self-shrinkers in $mathbb{R}^{n+1}$ with scalar curvature bounded from above or below by some constant are isometric to the round sphere $mathbb{S}^n(sqrt{n})$, which implies that $n$-dimensional compact self-shrinkers in $mathbb{R}^{n+1}$ with constant scalar curvature are isometric to the round sphere $mathbb{S}^n(sqrt{n})$(see also cite{Hui1}). Complete classifications of $n$-dimensional translating solitons in $mathbb{R}^{n+1}$ with nonnegative constant scalar curvature and of $n$-dimensional self-expanders in $mathbb{R}^{n+1}$ with nonnegative constant scalar curvature were given by Mart{i}n, Savas-Halilaj and Smoczykcite{MSS} and Ancari and Chengcite{AC}, respectively. In this paper we give complete classifications of $n$-dimensional complete self-shrinkers in $mathbb{R}^{n+1}$ with nonnegative constant scalar curvature. We will also give alternative proofs of the classification theorems due to Mart{i}n, Savas-Halilaj and Smoczyk cite{MSS} and Ancari and Chengcite{AC}.
We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an $L^2$ bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee, and M.-P. Tsui, shows that these conditions are optimal.
By studying the monotonicity of the first nonzero eigenvalues of Laplace and p-Laplace operators on a closed convex hypersurface $M^n$ which evolves under inverse mean curvature flow in $mathbb{R}^{n+1}$, the isoperimetric lower bounds for both eigenvalues were founded.
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