Capillary surfaces of constant mean curvature in a right solid cylinder

In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the solid cylinder, we obtain some results of symmetry by using the Alexandrov reflection method. When the mean curvature is zero, we give sufficient conditions to conclude that the surface is part of a plane or a catenoid.     

Extrinsic curvature of semiconvex subspaces in Alexandrov geometry

In Alexandrov spaces of curvature bounded either above (CBA) or below (CBB), we obtain extrinsic curvature bounds on subspaces associated with semiconcave functions. These subspaces play the role in singular geometry of submanifolds in Riemannian geometry, and arise naturally in many different places. For CBA spaces, we obtain new intrinsic curvature bounds on subspaces. For CBB spaces whose boundary is extrinsically curved, we strengthen Perelman’s concavity theorem for distance from the boundary, deriving corollaries on sharp diameter bounds, contractibility, and rigidity.     

A splitting theorem for infinite dimensional Alexandrov spaces with nonnegative curvature and its applications

We prove a splitting theorem for Alexandrov space of nonnegative curvature without properness assumption. As a corollary, we obtain a maximal radius theorem for Alexandrov spaces of curvature bounded from below by 1 without properness assumption. Also, we provide new examples of infinite dimensional Alexandrov spaces of nonnegative curvature.     

A quadrangle comparison theorem and its application to soul theory for Alexandrov spaces

We shall derive two sufficient conditions for complete finite-dimensional Alexandrov spaces of nonnegative curvature to be contractible. One of the new technical tools used in our proof is a quadrangle comparison theorem inspired by Perelman.     

Concepts of curvatures in normed planes

The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curvature concepts, we try to fill this gap. To complete and clarify the whole picture, we show which known concepts are equivalent, and add also a new type of curvature. Certainly, this yields a basis for further research and also for possible extensions of the whole existing framework. In addition, we derive various new results referring in full broadness to the variety of known curvature types in normed planes. These new results involve characterizations of curves of constant curvature, new characterizations of Radon planes and the Euclidean subcase, and analogues to classical statements like the four vertex theorem and the fundamental theorem on planar curves. We also introduce a new curvature type, for which we verify corresponding properties. As applications of the little theory developed in our expository paper, we study the curvature behavior of curves of constant width and obtain also new results on notions like evolutes, involutes, and parallel curves.     

Parallel Transportation for Alexandrov Space with Curvature Bounded Below

In this paper we construct a "synthetic" parallel transportation along a geodesic in Alexandrov space with curvature bounded below, and prove an analog of the second variation formula for this case. A closely related construction has been made for Alexandrov space with bilaterally bounded curvature by Igor Nikolaev (see [N]).?Naturally, as we have a more general situation, the constructed transportation does not have such good properties as in the case of bilaterally bounded curvature. In particular, we cannot prove the uniqueness in any good sense. Nevertheless the constructed transportation is enough for the most important applications such as Synge's lemma and Frankel's theorem. Recently by using this parallel transportation together with techniques of harmonic functions on Alexandrov space, we have proved an isoperimetric inequality of Gromov's type.?Author is indebted to Stephanie Alexander, Yuri Burago and Grisha Perelman for their willingness to understand, interest and important remarks. Submitted: January 1997, Revised version: June 1997     

Infinite curvature on typical convex surfaces

Solving a long-standing open question of Zamfirescu, we will show that typical convex surfaces contain points of infinite curvature in all tangent directions. To prove this, we use an easy curvature definition imitating the idea of Alexandrov spaces of bounded curvature, and show continuity properties for this notion.     

A Geometric Problem and the Hopf Lemma. Ⅱ

A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in Rn+1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn+1 =constant in case M satisfies: for any two points (X′, Xn+1), (X′, ^Xn+1)on M, with Xn+1 ＞ ^Hn+1, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part Ⅰ dealt with corresponding one dimensional problems.     

Complete embedded minimal surfaces of finite total curvature with planar ends of smallest possible order

Using a method by Traizet (J Differ Geom 60:103–153, 2002), which reduces the construction of minimal surfaces via the Weierstraß Theorem and the implicit function theorem to solving algebraic equations in several complex variables, we will show the existence of complete embedded minimal surfaces of finite total curvature with planar ends of least possible order.     

Symmetry of solitary waves

It is shown that all supercritical solitary wave solutions to the equations for water waves are symmetric, and monotone on either side of the crest. The proof is based on the Alexandrov method of moving planes. Further a priori estimates, and asymptotic decay properties of solutions are derived     

Locally symmetric minimal affine Lagrangian surfaces in C2

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C ² must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection. In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds. In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C ². Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry.     

Mechanical theorem proving in the surfaces using the characteristic set method and Wronskian determinant

In this paper, we generalize the method of mechanical theorem proving in curves to prove theorems about surfaces in differential geometry with a mechanical procedure. We improve the classical result on Wronskian determinant, which can be used to decide whether the elements in a partial differential field are linearly dependent over its constant field. Based on Wronskian determinant, we can describe the geometry statements in the surfaces by an algebraic language and then prove them by the characteristic set method.     

Estimation of the Visibility Angle of a Curve in a Metric Space of Nonpositive Curvature

We give an elementary proof to the inequality estimating some characteristic of a curve, the visibility angle of the curve from a given point, through the integral curvature of the curve. We consider the case of curves in a metric space of nonpositive curvature in the sense of Alexandrov.     

Locally symmetric minimal affine Lagrangian surfaces in C<Superscript>2</Superscript>

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C ² must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection. In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds. In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C ². Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry. The research supported by the KBN grant 1 PO3A 034 26.     

Kirszbraun's Theorem and Metric Spaces of Bounded Curvature

We generalize Kirszbraun's extension theorem for Lipschitz maps between (subsets of) euclidean spaces to metric spaces with upper or lower curvature bounds in the sense of A.D. Alexandrov. As a by-product we develop new tools in the theory of tangent cones of these spaces and obtain new characterization results which may be of independent interest. Submitted: June 1996, final version: November 1996     

关于曲率积分不等式的注记

本文主要研究平面卵形线的曲率积分不等式.利用积分几何中凸集的支持函数以及外平行集的性质,得到了Gage等周不等式与曲率的熵不等式的一个积分几何的简化证明;进一步地,我们得到了一个新的关于曲率积分的不等式.     

Convex functions on Alexandrov surfaces

We investigate the topological structure of Alexandrov surfaces of curvature bounded below which possess convex functions. We do not assume the continuities of these functions. Nevertheless, if the convex functions satisfy a condition of local nonconstancy, then the topological structures of Alexandrov surfaces and the level sets configurations of these functions in question are determined.

     

Connections in fiberings associated with the Grassman manifold and the space of centered planes

In the present paper we study connections in the fiberings associated with the Grassmann manifold and the space of the centered planes. The work is related to the studies in differential geometry. In the paper, we use the method of continuations and scopes of G. F. Laptev which generalizes the moving frame method and the exterior forms method of Cartan; the method depends on calculation of exterior differential forms. In the paper, we develop a new method of research in Grassman manifolds and some generalization of the method which includes the theory of the induced connections of the spaces of planes and centered planes in the n-dimensional projective space.     

Qualitative properties of blow-up solutions to some semilinear elliptic systems in non-convex domains

In this paper we study qualitative properties of boundary blow-up solutions to some semilinear elliptic cooperative systems in bounded non-convex domains. In particular, by a careful adaptation of the celebrated moving plane procedure of Alexandrov–Serrin, we deduce symmetry and monotonicity results for blow-up solutions for this class of systems.     

Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces

We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.