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Register a new userOn minimal surfaces immersed in three dimensional Kropina Minkowski space
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In this paper we consider a three dimensional Kropina space and obtain the partial differential equation that characterizes a minimal surfaces with the induced metric. Using this characterization equation we study various immersions of minimal surfaces. In particular, we obtain the partial differential equation that characterizes the minimal translation surfaces and show that the plane is the only such surface.
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In this paper we consider the Matsumoto metric $F=frac{alpha^2}{alpha-beta}$, on the three dimensional real vector space and obtain the partial differential equations that characterize the minimal surfaces which are graphs of smooth functions and then we prove that plane is the only such surface. We also obtain the partial differential equation that characterizes the minimal translation surfaces and show that again plane is the only such surface.
We prove that under some assumptions on the mean curvature the set of umbilical points of an immersed surface in a $3$-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of the celebrated Hopf theorem.
In this paper, we introduce two notions on a surface in a contact manifold. The first one is called degree of transversality (DOT) which measures the transversality between the tangent spaces of a surface and the contact planes. The second quantity, called curvature of transversality (COT), is designed to give a comparison principle for DOT along characteristic curves under bounds on COT. In particular, this gives estimates on lengths of characteristic curves assuming COT is bounded below by a positive constant. We show that surfaces with constant COT exist and we classify all graphs in the Heisenberg group with vanishing COT. This is accomplished by showing that the equation for graphs with zero COT can be decomposed into two first order PDEs, one of which is the backward invisicid Burgers equation. Finally we show that the p-minimal graph equation in the Heisenberg group also has such a decomposition. Moreover, we can use this decomposition to write down an explicit formula of a solution near a regular point.
Concerning the value distribution problem for generalized Gauss maps, we not only generalize Fujimotos theorem to complete space-like stationary surfaces in Minkowski spacetime, but also estimate the upper bound of the number of exceptional values when the Gauss image lies in the graph of a rational function f of degree m, showing a sharp contrast to Bernstein type results for minimal surfaces in 4-dimensional Euclidean space. Moreover, we introduce the conception of conjugate similarity on the special linear group to classify all degenerate stationary surfaces (i.e. m=0 or 1), and establish several structure theorems for complete stationary graphs in Minkowski spacetime from the viewpoint of the degeneracy of Gauss maps.
We prove that if $phi colon mathbb{R}^2 to mathbb{R}^{1+2}$ is a smooth proper timelike immersion with vanishing mean curvature, then necessarily $phi$ is an embedding, and every compact subset of $phi(mathbb{R}^2)$ is a smooth graph. It follows that if one evolves any smooth self-intersecting spacelike curve (or any planar spacelike curve whose unit tangent vector spans a closed semi-circle) so as to trace a timelike surface of vanishing mean curvature in $mathbb{R}^{1+2}$, then the evolving surface will either fail to remain timelike, or it will fail to remain smooth. We show that, even allowing for null points, such a Cauchy evolution must undergo a scalar curvature blow-up---where the blow-up is with respect to an $L^1L^infty$ norm---and thus the evolving surface will be $C^2$ inextendible beyond singular time. In addition we study the continuity of the unit tangent for the evolution of a self-intersecting curve in isothermal gauge, which defines a well-known evolution beyond singular time.
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