In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $mathbb{R}^{n+1}$ with speed $u^alpha f^{-beta}$, where $u$ is the support function of the hypersurface, $f$ is a smooth, symme
tric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. For $alpha le 0<betale 1-alpha$, we prove that the flow has a unique smooth solution for all time, and converges smoothly after normalization, to a sphere centered at the origin. In particular, the results of Gerhardt cite{GC3} and Urbas cite{UJ2} can be recovered by putting $alpha=0$ and $beta=1$ in our first result. If the initial hypersurface is convex, this is our previous work cite{DL}. If $alpha le 0<beta< 1-alpha$ and the ambient space is hyperbolic space $mathbb{H}^{n+1}$, we prove that the flow $frac{partial X}{partial t}=(u^alpha f^{-beta}-eta u) u$ has a longtime existence and smooth convergence to a coordinate slice. The flow in $mathbb{H}^{n+1}$ is equivalent (up to an isomorphism) to a re-parametrization of the original flow in $mathbb{R}^{n+1}$ case. Finally, we find a family of monotone quantities along the flows in $mathbb{R}^{n+1}$. As applications, we give a new proof of a family of inequalities involving the weighted integral of $k$th elementary symmetric function for $k$-convex, star-shaped hypersurfaces, which is an extension of the quermassintegral inequalities in cite{GL2}.
Presenting high-resolution (HR) human appearance is always critical for the human-centric videos. However, current imagery equipment can hardly capture HR details all the time. Existing super-resolution algorithms barely mitigate the problem by only
considering universal and low-level priors of im-age patches. In contrast, our algorithm is under bias towards the human body super-resolution by taking advantage of high-level prior defined by HR human appearance. Firstly, a motion analysis module extracts inherent motion pattern from the HR reference video to refine the pose estimation of the low-resolution (LR) sequence. Furthermore, a human body reconstruction module maps the HR texture in the reference frames onto a 3D mesh model. Consequently, the input LR videos get super-resolved HR human sequences are generated conditioned on the original LR videos as well as few HR reference frames. Experiments on an existing dataset and real-world data captured by hybrid cameras show that our approach generates superior visual quality of human body compared with the traditional method.
In this paper, we consider an expanding flow of closed, smooth, uniformly convex hypersurface in Euclidean mathbb{R}^{n+1} with speed u^alpha f^beta (alpha, betainmathbb{R}^1), where u is support function of the hypersurface, f is a smooth, symmetric
, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If alpha leq 0<betaleq 1-alpha, we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalization, to a round sphere centered at the origin.
This paper concerns the evolution of complete noncompact locally uniformly convex hypersurface in Euclidean space by curvature flow, for which the normal speed $Phi$ is given by a power $betageq 1$ of a monotone symmetric and homogeneous of degree on
e function $F$ of the principal curvatures. Under the assumption that $F$ is inverse concave and its dual function approaches zero on the boundary of positive cone, we prove that the complete smooth strictly convex solution exists and remains a graph until the maximal time of existence. In particular, if $F=K^{s/n}G^{1-s}$ for any $sin(0, 1]$, where $G$ is a homogeneous of degree one, increasing in each argument and inverse concave curvature function, we prove that the complete noncompact smooth strictly convex solution exists and remains a graph for all times.
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.
This paper concerns closed hypersurfaces of dimension $n(geq 2)$ in the hyperbolic space ${mathbb{H}}_{kappa}^{n+1}$ of constant sectional curvature $kappa$ evolving in direction of its normal vector, where the speed is given by a power $beta (geq 1/
m)$ of the $m$th mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the $mbox{Gauss}$ curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvature is close enough to 1 everywhere, depending only on $n$, $m$, $beta$ and $kappa$, then under the flow this is maintained, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces exponentially converge to a geodesic sphere of ${mathbb{H}}_{kappa}^{n+1}$, enclosing the same volume as the initial hypersurface.
We prove the mean curvature flow of a spacelike graph in $(Sigma_1times Sigma_2, g_1-g_2)$ of a map $f:Sigma_1to Sigma_2$ from a closed Riemannian manifold $(Sigma_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(Sigma_2,g_2)$ with bound
ed curvature tensor and derivatives, and with sectional curvatures satisfying $K_2leq K_1$, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption $K_2leq K_1$, that if $K_1>0$, or if $Ricci_1>0$ and $K_2leq -c$, $c>0$ constant, any map $f:Sigma_1to Sigma_2$ is trivially homotopic provided $f^*g_2<rho g_1$ where $rho=min_{Sigma_1}K_1/sup_{Sigma_2}K_2^+geq 0$, in case $K_1>0$, and $rho=+infty$ in case $K_2leq 0$. This largely extends some known results for $K_i$ constant and $Sigma_2$ compact, obtained using the Riemannian structure of $Sigma_1times Sigma_2$, and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo-Riemannian setting then in the Riemannian one.
We generalize a Bernstein-type result due to Albujer and Alias, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $Sigma_1times mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete spacelike graphic sub
manifold with parallel mean curvature, defined by a map $f: Sigma_1to Sigma_2$ between two Riemannian manifolds $(Sigma_1^m, g_1)$ and $(Sigma^n_2, g_2)$ of sectional curvatures $K_1$ and $K_2$, respectively. We take on $Sigma_1times Sigma_2$ the pseudo-Riemannian product metric $g_1-g_2$. Under the curvature conditions, $mathrm{Ricci}_1 geq 0$ and $K_1geq K_2$, we prove that, if the second fundamental form of $M$ satisfies an integrability condition, then $M$ is totally geodesic, and it is a slice if $mathrm{Ricci}_1(p)>0$ at some point. For bounded $K_1$, $K_2$ and hyperbolic angle $theta$, we conclude $M$ must be maximal. If $M$ is a maximal surface and $K_1geq K_2^+$, we show $M$ is totally geodesic with no need for further assumptions. Furthermore, $M$ is a slice if at some point $pin Sigma_1$, $K_1(p)> 0$, and if $Sigma_1$ is flat and $K_2<0$ at some point $f(p)$, then the image of $f$ lies on a geodesic of $Sigma_2$.
Given $(bar{M},Omega)$ a calibrated Riemannian manifold with a parallel calibration of rank $m$, and $M^m$ an immersed orientable submanifold with parallel mean curvature $H$ we prove that if $cos theta$ is bounded away from zero, where $theta$ is th
e $Omega$-angle of $M$, and if $M$ has zero Cheeger constant, then $M$ is minimal. In the particular case $M$ is complete with $Ricc^Mgeq 0$ we may replace the boundedness condition on $cos theta$ by $cos thetageq Cr^{-beta}$, when $rto +infty$, where $ 0leqbeta <1 $ and $C > 0$ are constants and $r$ is the distance function to a point in $M$. Our proof is surprisingly simple and extends to a very large class of submanifolds in calibrated manifolds, in a unified way, the problem started by Heinz and Chern of estimating the mean curvature of graphic hypersurfaces in Euclidean spaces. It is based on a estimation of $|H|$ in terms of $costheta$ and an isoperimetric inequality. We also give some conditions to conclude $M$ is totally geodesic. We study some particular cases.
This is a survey of our work on spacelike graphic submanifolds in pseudo-Riemannian products, namely on Heinz-Chern and Bernstein-Calabi results and on the mean curvature flow, with applications to the homotopy of maps between Riemannian manifolds.
