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Register a new userMinimal surfaces in three-dimensional Matsumoto space
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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.
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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.
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.
It is known that minimal surfaces in Euclidean space can be represented in terms of holomorphic functions. For example, we have the well-known Weierstrass representation, where part of the holomorphic data is chosen to be the stereographic projection of the normal of the corresponding surface, and also the Bjorling representation, where it is prescribed a curve on the surface and the unit normal on this curve. In this work, we are interested in the holomorphic representation of minimal surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the isotropic metric is degenerate, a surface normal cannot be unequivocally defined based on metric properties only, which leads to distinct definitions of an isotropic normal. As a consequence, this may also lead to distinct forms of a Weierstrass and of a Bjorling representation. Here, we show how to represent simply isotropic minimal surfaces in accordance with the choice of an isotropic surface normal.
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.
We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $mathbb{R}^{3}_{raisepunct{.}}$ We also show that any minimal hypersurface immersed with bounded curvature in $Mtimes R_+$ equals some $Mtimes {s}$ provided $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with $text{Ric}_M geq 0$ and whose sectional curvatures are bounded from above. For $H$-surfaces we prove that a stochastically complete surface $M$ can not be in the mean convex side of a $H$-surface $N$ embedded in $R^3$ with bounded curvature if $sup vert H_{_M}vert < H$, or ${rm dist}(M,N)=0$ when $sup vert H_{_M}vert = H$. Finally, a maximum principle at infinity is shown assuming $M$ has non-empty boundary.
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