Related papers: A unified flow approach to smooth, even $L_p$-Mink…
In this paper, we investigate an $L_{p}$ Christoffel-Minkowski-type problem that prescribes a class of $L_p$ geometric measures, which are mixtures of the $k$-th area measure and the $q$-th dual curvature measure. By establishing a gradient…
We obtain height, gradient, and curvature a priori estimates for a modified mean curvature flow in Riemannian manifolds endowed with a Killing vector field. As a consequence, we prove the existence of smooth, entire, longtime solutions for…
We study the motion of a droplet evolving by mean curvature with volume constraint and contact angle condition on a half space. We prove the existence of a global-in-time weak solution, called the flat flow. A difficulty arises when we…
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…
We present algorithms for solving a large class of flow and regression problems on unit weighted graphs to $(1 + 1 / poly(n))$ accuracy in almost-linear time. These problems include $\ell_p$-norm minimizing flow for $p$ large ($p \in…
In this paper we study a flow by minkowskian curvature where we have a different Minkowski plane at each time. We derive some evolution formulas, present sufficient hypotesis for the short time existence and convexity of solutions and study…
We study a notion of relative entropy motivated by self-expanders of mean curvature flow. In particular, we obtain the existence of this quantity for arbitrary hypersurfaces trapped between two disjoint self-expanders asymptotic to the same…
We investigate the existence of weak expanding solutions of the harmonic map flow for maps with values into a smooth closed Riemannian manifold. We prove the existence of such solutions in case the target manifold is isometrically embedded…
We construct smooth mean curvature flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and…
This article surveys recent progress of results in topology and dynamics based on techniques of closed one-forms. Our approach allows us to draw conclusions about properties of flows by studying homotopical and cohomological features of…
In this paper, for a given compact 3-manifold with an initial Riemannian metric and a symmetric tensor, we establish the short-time existence and uniqueness theorem for extension of cross curvature flow. We give an example of this flow on…
In this paper, we prove that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. The main…
We show that self-similar solutions for the mean curvature flow, surface diffusion and Willmore flow of entire graphs are stable upon perturbations of initial data with small Lipschitz norm. Roughly speaking, the perturbed solutions are…
We prove several sharp one-sided pinching estimates for immersed and embedded hypersurfaces evolving by various fully nonlinear, one-homogeneous curvature flows by the method of Stampacchia iteration. These include sharp estimates for the…
The dual Minkowski problem in the two-dimensional plane is studied in this paper. By combining the theoretical analysis and numerical estimation of an integral with parameters, we find the number of solutions to this problem for the…
In this paper, by using new auxiliary functions, we study a class of contracting flows of closed, star-shaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^{\frac{\alpha}{\beta}}\sigma_k^{\frac{1}{\beta}}$, where $\sigma_k$ is the…
Necessary and sufficient conditions for the existence of solutions to the asymmetric $L_p$ Minkowski problem in $\mathbb{R}^2$ are established for $0 < p < 1$.
We present a numerical investigation of the evolution of the Hawking mass for perturbed surfaces evolving under hypersurface-restricted uniformly expanding flows in Minkowski spacetime. Although monotonicity of the Hawking mass under…
We prove three results in this paper. First, we prove for a wide class of functions $\varphi\in C^2(\mathbb{S}^{n-1})$ and $\psi(X, \nu)\in C^2(\mathbb{R}^{n+1}\times\mathbb{H}^n),$ there exists a unique, entire, strictly convex, spacelike…
Let X be a smooth subvariety of CP^N. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X, which attempts to deform the given embedding into a balanced one. If L->X is an ample line bundle,…