Compactness for immersions of prescribed Gaussian curvature II: geometric aspects

We develop a compactness result near the boundary for families of locally convex immersions. We also develop a mod 2 degree theory for immersion of constant (and prescribed) Gaussian curvature with prescribed boundary. These are then used to solve the Plateau problem for immersions of constant (and prescribed) Gaussian curvature in general Hadamard manifolds.     

Gaussian curvature estimates for the convex level sets of p‐harmonic functions

We give a positive lower bound for the Gaussian curvature of the convex level sets of p‐harmonic functions with the Gaussian curvature of the boundary and the norm of the gradient on the boundary. Combining the deformation process, this estimate gives a new approach to studying the convexity of the level sets of the p‐harmonic function. © 2010 Wiley Periodicals, Inc.     

The Convexity and the Gaussian Curvature Estimates for the Level Sets of Harmonic Functions on Convex Rings in Space Forms

In this paper, we first establish a constant rank theorem for the second fundamental form of the convex level sets of harmonic functions in space forms. Applying the deformation process, we prove that the level sets of the harmonic functions on convex rings in space forms are strictly convex. Moreover, we give a lower bound for the Gaussian curvature of the convex level sets of harmonic functions in terms of the Gaussian curvature of the boundary and the norm of the gradient on the boundary.     

Some sufficient conditions for immersion in the large of metrics of positive curvature

It is proved that a regular Riemannian manifold diffeomorphic to a circle and having positive Gaussian curvature bounded from zero is immersible into a three-dimensional Euclidean space in the form of a regular surface if it has smallL _p (the norm of the gradient of Gaussian curvature), p > 2, or if it has a sufficiently small area (with any behavior of the geodesic boundary curvature).Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 51–59, 1991.     

The Ricci flow on the two ball with a rotationally symmetric metric

In this paper we study a boundary-value problem for the Ricci flow in the two-dimensional ball endowed with a rotationally symmetric metric of positive Gaussian curvature and prove short and long time existence results. We construct families of metrics for which the flow uniformizes the curvature along a sequence of times. Finally, we show that if the initial metric has positive Gaussian curvature and the boundary has positive geodesic curvature then the flow uniformizes the curvature along a sequence of times. The text was submitted by the author in English.     

Lattice Points in Three-Dimensional Convex Bodies with Points of Gaussian Curvature Zero at the Boundary

 In this article we investigate the number of lattice points in a three-dimensional convex body which contains non-isolated points with Gaussian curvature zero but a finite number of flat points at the boundary. Especially, in case of rational tangential planes in these points we investigate not only the influence of the flat points but also of the other points with Gaussian curvature zero on the estimation of the lattice rest.     

Convexity estimates for the solutions of a class of semi-linear elliptic equations

In this paper, we are concerned with convexity estimates for solutions of a class of semi-linear elliptic equations involving the Laplacian with power-type nonlinearities. We consider auxiliary curvature functions which attain their minimum values on the boundary and then establish lower bound convexity estimates for the solutions. Then we give two applications of these convexity estimates. We use the deformation method to prove a theorem concerning the strictly power concavity properties of the smooth solutions to these semi-linear elliptic equations. Finally, we give a sharp lower bound estimate of the Gaussian curvature for the solution surface of some specific equation by the curvatures of the domain's boundary.     

On p-Willmore disks with boundary energies

We consider an energy functional on surface immersions which includes contributions from both boundary and interior. Inspired by physical examples, the boundary is modeled as the center line of a generalized Kirchhoff elastic rod, while the interior term is arbitrarily dependent on the mean curvature and linearly dependent on the Gaussian curvature. We study equilibrium configurations for this energy in general among topological disks, as well as specifically for the class of examples known as p-Willmore energies.     

Shape-dependent natural boundary condition of Lagrangian field

In the framework of the Lagrangian field theory, we derive the equations characterizing shape-dependent natural boundary conditions from the Hamilton’s principle. Of these equations, one exhibits mathematical pattern similar to general relativity. In this equation, one side of the sign of equality is the energy–momentum tensor of field and another side is the combination of mean curvature and Gaussian curvature of boundary surface. Meanwhile, we verify that the shape-dependent natural boundary condition can be simplified into the shape equation of lipid vesicle or the generalized Young–Laplace’s equation under different condition.     

The isoperimetric inequality on a surface

We prove a new isoperimetric inequality which relates the area of a multiply connected curved surface, its Euler characteristic, the length of its boundary, and its Gaussian curvature. Received: 31 July 1998     

Compact Surfaces with Constant Gaussian Curvature in Product Spaces

We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.     

Stability and low-frequency oscillations of thin shells with entirely or partially negative Gaussian curvature

The paper is concerned with determination of the lower part of the spectrum of shells of revolution of entirely or partially negative Gaussian curvature. A classification of the integrals of the system of equations is obtained in terms of the geometry of the middle surface and boundary conditions. Special attention has been paid to the problem of turning point, at which the curvature changes its sign.     

Lattice Points in Three-Dimensional Convex Bodies with Points of Gaussian Curvature Zero at the Boundary

 In this article we investigate the number of lattice points in a three-dimensional convex body which contains non-isolated points with Gaussian curvature zero but a finite number of flat points at the boundary. Especially, in case of rational tangential planes in these points we investigate not only the influence of the flat points but also of the other points with Gaussian curvature zero on the estimation of the lattice rest. Received 19 June 2001; in revised form 17 January 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Ruled surfaces with vanishing second Gaussian curvature

For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid.     

Nirenberg’s problem on domains in the 2-sphere

In this article, we consider the problem of prescribing Gaussian curvature on domains in the unit 2-sphere. We obtain the existence result for any domain with area between (2π, 4π) and having at least 2 boundary components.     

Korn inequalities for shells with zero Gaussian curvature

We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.     

A Weierstrass representation for linear Weingarten spacelike surfaces of maximal type in the Lorentz-Minkowski space

In this work we extend the Weierstrass representation for maximal spacelike surfaces in the 3-dimensional Lorentz-Minkowski space to spacelike surfaces whose mean curvature is proportional to its Gaussian curvature (linear Weingarten surfaces of maximal type). We use this representation in order to study the Gaussian curvature and the Gauss map of such surfaces when the immersion is complete, proving that the surface is a plane or the supremum of its Gaussian curvature is a negative constant and its Gauss map is a diffeomorphism onto the hyperbolic plane. Finally, we classify the rotation linear Weingarten surfaces of maximal type.     

关于超曲面上测度的Fourier变换的零点

设Ｓ是Ｅｕｃｌｉｄ空间Ｒ￣ｎ（ｎ≥２）中一个紧闭光滑超曲面，它关于原点中心对称，其Ｇａｕｓｓ曲率处处非零。设ｄμ是Ｓ上一个光滑正测度，是其Ｆｏｕｒｉｅｒ变换。本文证明，的零点集是一个紧集与可列多个微分同胚于单位球面的超曲面之无交并．     

Translating Surfaces of the Non-parametric Mean Curvature Flow in Lorentz Manifold M2×R

In this paper, for the Lorentz manifold M²× R with M² a 2-dimensional complete surface with nonnegative Gaussian curvature, the authors investigate its spacelike graphs over compact, strictly convex domains in M², which are evolving by the nonparametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.     

Lattice Points in Convex Bodies with Planar Points on the Boundary

 An asymptotic formula is proved for the number of lattice points in large threedimensional convex bodies. In contrast to the usual assumption the Gaussian curvature of the boundary may vanish at non-isolated points. It is only assumed that the second fundamental form vanishes at isolated points where the tangent plane is rational and some ellipticity condition holds.