Isometric deformation of surfaces in Kahler plane preserving the mean curvature function

In the ambient space of a semidirect product $\mathbb{R}^{2}\rtimes _{A}\mathbb{R}$, we consider a connected domain ${\rm\Omega}\subseteq \mathbb{R}^{2}\rtimes _{A}\{0\}$. Given a function $u:{\rm\Omega}\rightarrow \mathbb{R}$, its ${\it\pi}$-graph is $\text{graph}(u)=\{(x,y,u(x,y))\mid (x,y,0)\in {\rm\Omega}\}$. In this paper we study the partial differential equation that $u$ must satisfy so that $\text{graph}(u)$ has prescribed mean curvature $H$. Using techniques from quasilinear elliptic equations we prove that if a ${\it\pi}$-graph has a nonnegative mean curvature function, then it satisfies some uniform height estimates that depend on ${\rm\Omega}$ and on the supremum the function attains on the boundary of ${\rm\Omega}$. When $\text{trace}(A)>0$, we prove that the oscillation of a minimal graph, assuming the same constant value $n$ along the boundary, tends to zero when $n\rightarrow +\infty$ and goes to $+\infty$ if $n\rightarrow -\infty$. Furthermore, we use these estimates, allied with techniques from Killing graphs, to prove the existence of minimal ${\it\pi}$-graphs assuming the value zero along a piecewise smooth curve ${\it\gamma}$ with endpoints $p_{1},\,p_{2}$ and having as boundary ${\it\gamma}\cup (\{p_{1}\}\times [0,\,+\infty ))\cup (\{p_{2}\}\times [0,\,+\infty ))$.

Download Full-text

Isometric Deformation of Surfaces in the Hyperbolic 3-Manifold Preserving the Mean Curvature

Tokyo Journal of Mathematics ◽

10.3836/tjm/1270043625 ◽

1995 ◽

Vol 18 (1) ◽

pp. 247-258 ◽

Cited By ~ 2

Author(s):

Hiroshi TAKEUCHI

Keyword(s):

Mean Curvature ◽

Isometric Deformation ◽

The Mean

Download Full-text

On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems

Canadian Journal of Mathematics ◽

10.4153/cjm-2000-024-x ◽

2000 ◽

Vol 52 (3) ◽

pp. 522-538 ◽

Cited By ~ 87

Author(s):

Changfeng Gui ◽

Juncheng Wei

Keyword(s):

Mean Curvature ◽

Variational Approach ◽

Existence Of Solutions ◽

Sphere Packing ◽

Curvature Function ◽

Bounded Smooth Domain ◽

Neumann Problems ◽

Subcritical Nonlinearity ◽

The Mean ◽

Bounded Smooth

AbstractWe consider the problemwhere Ω is a bounded smooth domain in RN, ε > 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses multiple boundary spike solutions that concentrate, as ε approaches zero, at multiple critical points of the mean curvature function H(P), P ∈ ∂Ω. It is also proved that this equation has multiple interior spike solutions which concentrate, as ε → 0, at sphere packing points in Ω.In this paper, we prove the existence of solutions with multiple spikes both on the boundary and in the interior. The main difficulty lies in the fact that the boundary spikes and the interior spikes usually have different scales of error estimation. We have to choose a special set of boundary spikes to match the scale of the interior spikes in a variational approach.

Download Full-text

Mean curvature and shape operator of isometric immersions in real-space-forms

Glasgow Mathematical Journal ◽

10.1017/s001708950003130x ◽

1996 ◽

Vol 38 (1) ◽

pp. 87-97 ◽

Cited By ~ 46

Author(s):

Bang-Yen Chen

Keyword(s):

Riemannian Manifold ◽

Euclidean Space ◽

Mean Curvature ◽

Shape Operator ◽

Real Space ◽

Curvature Function ◽

Isometric Immersions ◽

Space Forms ◽

Real Space Forms ◽

The Mean

According to the well-known Nash's theorem, every Riemannian n-manifold admits an isometric immersion into the Euclidean space En(n+1)(3n+11)/2. In general, there exist enormously many isometric immersions from a Riemannian manifold into Euclidean spaces if no restriction on the codimension is made. For a submanifold of a Riemannian manifold there are associated several extrinsic invariants beside its intrinsic invariants. Among the extrinsic invariants, the mean curvature function and shape operator are the most fundamental ones.

Download Full-text

A Class of Weingarten Surfaces in Euclidean 3-Space

Abstract and Applied Analysis ◽

10.1155/2013/398158 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Yu Fu ◽

Lan Li

Keyword(s):

Mean Curvature ◽

Complete Classification ◽

Curvature Function ◽

Weingarten Surfaces ◽

The Mean

The class of biconservative surfaces in Euclidean 3-space𝔼3are defined in (Caddeo et al., 2012) by the equationA(grad H)=-H grad Hfor the mean curvature functionHand the Weingarten operatorA. In this paper, we consider the more general case that surfaces in𝔼3satisfyingA(grad H)=kH grad Hfor some constantkare called generalized bi-conservative surfaces. We show that this class of surfaces are linear Weingarten surfaces. We also give a complete classification of generalized bi-conservative surfaces in𝔼3.

Download Full-text

Pansu–Wulff shapes in ℍ1

Advances in Calculus of Variations ◽

10.1515/acv-2020-0093 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Julián Pozuelo ◽

Manuel Ritoré

Keyword(s):

Mean Curvature ◽

Isoperimetric Problem ◽

Variational Formula ◽

Geodesic Curvature ◽

Vertical Axis ◽

Singular Part ◽

Tangent Plane ◽

Curvature Function ◽

Invariant Norm ◽

The First Variation

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

Download Full-text

Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

Open Mathematics ◽

10.1515/math-2020-0090 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1518-1530

Author(s):

Xuesen Qi ◽

Ximin Liu

Keyword(s):

Laplace Operator ◽

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow ◽

Second Fundamental Form ◽

Coefficient Function ◽

First Eigenvalue ◽

Initial Hypersurface ◽

The Mean ◽

The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

Download Full-text