相关论文: Interior Gradient Estimates for Anisotropic Mean C…
In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional $E$. We show…
In this work, we study graphs in $\M^n\times\Real$ that are evolving by the mean curvature flow over a bounded domain on $\M^n$, with prescribed contact angle in the boundary. We prove that solutions converge to translating surfaces in…
We consider the evolution by mean curvature of smooth $n$-dimensional submanifolds in $\mathbb{R}^{n+k}$ which are compact and quadratically pinched. We will be primarily interested in flows of high codimension, the case $k\geq 2$. We prove…
In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to mean curvature flow as well as the nonlinear flow by powers of the mean curvature.
Mapping complex input data into suitable lower dimensional manifolds is a common procedure in machine learning. This step is beneficial mainly for two reasons: (1) it reduces the data dimensionality and (2) it provides a new data…
In this paper we obtain a sharp height estimate concerning compact hypersurfaces immersed into warped product spaces with some constant higher order mean curvature, and whose boundary is contained into a slice. We apply these results to…
Extending the earlier results for analytic curve segments, in this article we describe the asymptotic behaviour of evolution of a finite segment of a C^n-smooth curve under the geodesic flow on the unit tangent bundle of a finite volume…
We consider a family of embedded, mean convex hypersurfaces in a Riemannian manifold which evolve by the mean curvature flow. We show that, given any number $T>0$ and any $\delta>0$, we can find a constant $C_0$ with the following property:…
We establish area bounds for two-dimensional immersions in R^3 and R^n. Namely, for \mu-stable immersions in R^3 (R^n), for graphs in $\mathbb R^3$ which solve quasilinear equations in divergence form, and for graphs which are critical for…
Since the seminal paper by Mitra et al., diffusion MR has been widely used in order to estimate surface-to-volume ratios. In the present work we generalize Mitra's formula for arbitrary diffusion encoding waveforms, including recently…
In the article, we generalize some recent results of Colding and Minicozzi on generic singularities of mean curvature flow to curved ambient spaces. To do so, we make use of a weighted monotonicity formula to derive an "almost monotonicity"…
We investigate the motion of a family of closed curves evolving according to the geometric evolution law on a given two dimensional manifold which is embedded or immersed in the three-dimensional Euclidean space. We derive a system of…
We consider the geometric evolution of a network in the plane, flowing by anisotropic curvature. We discuss local existence of a classical solution in the presence of several smooth anisotropies. Next, we discuss some aspects of the…
This note concerns the area growth and bottom spectrum of complete stable minimal surfaces in a three-dimensional manifold with scalar curvature bounded from below. When the ambient manifold is the Euclidean space, by an elementary…
In this paper, we prove a total curvature estimate of closed hypersurfaces in simply-connected non-positively curved symmetric spaces, and as a corollary, we obtain an isoperimetric inequality for such manifolds.
We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained…
In this work the evolution of a fluid droplet in vacuum is considered. This means that the surface tension and the fluid forces are in equilibrium at the free boundary. The fluid is governed by the incompressible quasi-steady Stokes…
We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in…
We explore the fundamental flow structure of inclined gravity currents with direct numerical simulations. A velocity maximum naturally divides the current into inner and outer shear layers, which are weakly coupled by exchange of momentum…
We report results of dissipative particle dynamics simulations and develop a semi-analytical theory of an anisotropic flow in a parallel-plate channel with two superhydrophobic striped walls. Our approach is valid for any local slip at the…