arXiv research

Gradient estimate for linearized translator equation in R^4.

The paper classifies noncollapsed translators in 4D space.

The isotropic 3-space \mathbb{I}^{3} is a real affine 3-space endowed with the metric dx^{2}+dy^{2}. In this paper we describe Weingarten and linear Weingarten affine translation surfaces in \mathbb{I}^{3}. Further we classify the affine translation surfaces in \mathbb{I}^{3} that satisfy certain equations in terms of …

The paper constructs hypersurfaces translating under powers of Gauss curvature.

Study on stability of 3D sessile drops, identifying degenerate kernel.

Study cohomological equation for robotic screw motions on SE(3).

We give an analytic approach to the translating soliton equation with a special emphasis in the study of the Dirichlet problem in convex domains of the plane.

Study left invariant spray geometry on Lie groups using parallel translations.

Study minimal surfaces in Kropina 3D space, finding only planes as minimal translation surfaces.

The paper studies complex curves with translation structures from differential equations.

The paper classifies shapes of translating solitons for a specific flow.

New methods prove existence of rotating shapes moving in space.

The paper classifies shapes of translating solitons from isoparametric graphs.

In this paper, we prove the existence of classical solutions of the Dirichlet problem for a class of quasi-linear elliptic equations on unbounded domains like a cone or a U-type domain. This problem comes from the study of mean curvature flow and its generalization, the flow by powers of mean curvature. Our approach is…

Classifies surfaces translating under specific curvature flows.

We extend the Mason-Newman Lax pair for the elliptic complex Monge-Ampère equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. We identify the real and imaginary parts of the potential, which we call partner symmetries, w…

Study classifies translators for mean curvature flow in 3D.

The paper classifies rotational K^α-translators in Minkowski space.

The paper classifies and describes translators in $SL(2,\mathbb{R})$ under specific symmetry conditions.

Paper analyzes blowup of regularized Jang solutions and constant expansion surfaces.

Study on Monge-Ampère equations with polynomial growth rates.

Study connects symmetries in dynamical systems to phase plane representations.

We prove the existence of classical solutions to the Dirichlet problem for the $α$-translating soliton equation defined in a strip of $\r^2$. We use the Perron method where a family of grim reapers are employed as barriers for solving the Dirichlet problem when the boundary data is formed by two copies of a convex func…

The height functions of K^(1/4)-flow translators in Euclidean space R^3 solve the unimodular Hessian equation. We explicitly and geometrically determine the moduli space of all helicoidal K^(1/4)-flow translators, which are generated from planar curves by the action of helicoidal groups.

We show that axisymmetric extremal horizons are unstable under linear scalar perturbations. Specifically, we show that translation invariant derivatives of generic solutions to the wave equation do not decay along such horizons as advanced time tends to infinity, and in fact, higher order derivatives blow up. This resu…

Random walks on hyperbolic spaces show linear growth in translation lengths.

A linear connection on a Finsler manifold is called compatible to the metric if its parallel transports preserve the Finslerian length of tangent vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a compatible linear connection. Since the compatibility to the Finslerian metric does not imply the…

Study translators in Generalised Robertson-Walker spacetimes, identifying warping functions and classifying examples.

Finite translation surfaces can be classified by the order of their singularities. When generalizing to infinite translation surfaces, however, the notion of order of a singularity is no longer well-defined and has to be replaced by new concepts. This article discusses the nature of two such concepts, recently introduc…

We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. We also give …

The paper proves Hessian estimates for specific geometric flows.

Study shows translators can have non-removable singularities at infinity but eventually converge to unique planes.

In this paper, we consider the problem of finding the hypersurface M^n in the Euclidean (n+1)-space R^{n+1} that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically characterizes the hypersurfaces in the upper halfspace (R^{n+1})_{+} with lowest gravi…

In many regular cases, there exists a (properly defined) limit of iterations of a function in several real variables, and this limit satisfies the functional equation (1-z)f(x)=f(f(xz)(1-z)/z); here z is a scalar and x is a vector. This is a special case of a well-known translation equation. In this paper we present a …

New translations defined; curve shortening flow solved in hyperbolic plane.

The study generalizes origamis to flat surfaces, exploring their combinatorial and geometric properties.

The paper explores how the Gauss curvature of Riemannian surfaces can be represented as the divergence of a vector field.

INO learns physical models with momentum conservation laws.

Construct intrinsic Langevin dynamics for rigid inclusions on curved surfaces.

A $λ$-translating soliton with density vector $\vec{v}$ is a surface in Euclidean space whose mean curvature $H$ satisfies $2H=2λ+\langle N,\vec{v}\rangle$, where $N$ is the Gauss map. We classify all $λ$-translating solitons that are invariant by a one-parameter group of translations and a one-parameter group of rotat…

We derive local $C^{2}$ estimates for complete non-compact translating solitons of the Gauss curvature flow in $\mathbb{R}^3$ which are graphs over a convex domain $Ω$. This is closely is related to deriving local $C^{1,1}$ estimates for the degenerate Monge-Ampére equation. As a result, given a weakly convex bounded d…

This is a review with examples concerning the concepts of affine (in particular, constant and linear) vector fields and fundamental vector fields on a manifold. The affine, linear and constant vector fields on a manifold are shown to be in a bijective correspondence with the fundamental vector fields on it of respectiv…

Optimal transport aligns source and target distributions for linear regression in 2D.

Study invariant $λ$-translators in Lorentz-Minkowski space.

UNSB uses neural Schrödinger Bridge to solve unpaired image-to-image translation.

Stability of catenoid in hyperbolic space proven without symmetry assumptions.

We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan-Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole. We find an une…

We construct solutions to the constraint equations in general relativity using the limit equation criterion introduced by Dahl, Humbert and the first author. We focus on solutions over compact 3-manifolds admitting a $\bS^1$-symmetry group. When the quotient manifold has genus greater than 2, we obtain strong far from …