On the scalar curvature of complex surfaces

Supported in part by NSF grant DMS-9204093.     

Completeness of curvature surfaces of submanifolds in complex space forms

We assume that on an open subset of a submanifold M of an arbitrary Riemannian ambient space N the eigenspaces of the shape operator of M induce a foliation L whose leaves are spherical submanifolds of N. In this situation we derive a condition which characterizes when the leaves of L are complete Riemannian submanifolds of M (see Theorem 2.4). We apply this result to real hypersurfaces of complex space forms, in particular Hopf hypersurfaces (see Theorem 3.2 and Proposition 3.3).     

Biminimal Lagrangian surfaces of constant mean curvature in complex space forms

We determine all biminimal Lagrangian surfaces of non-zero constant mean curvature in 2-dimensional complex space forms.     

Certain properties of surfaces of negative external curvature in Lobachevskian space

Several relationships are established bearing on the external curvature, K_e of surfaces in three-dimensional Lobachevskian space L₃ (K_e<0).Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 165–168, August, 1968.     

Some dynamic properties of volume preserving curvature driven flows

This paper is concerned with the following three types of geometric evolution equations: the volume preserving mean curvature flow, the intermediate surface diffusion flow, and the surface diffusion flow. Important common properties of these flows are the preservation of volume and the decrease of perimeter. It is shown in this paper that the intermediate surface diffusion flow can lose convexity. Hence the volume preserving mean curvature flow is the only flow among the evolution equations under consideration which preserves convexity, cf. [11, 16, 14, 17]. Moreover, several sufficient conditions are presented, which illustrate that each of the above mentioned flows can move smooth initial configurations into singularities in finite time.     

Some topological properties of cohomogeneity one manifolds with negative curvature

This paper is aimed at studying negatively curved Riemannian manifolds acted on by a Lie group of isometries with principal orbits of codimension one. The orbit space of such a manifold M is proved to be always homeomorphic to or ⁺ and this second case may occur only when either the singular orbit is a geodesic of M or when the space is simply connected. Several corollaries are given.     

Some curvature properties of Cartan spaces with mth root metrics

In this paper, we define some non-Riemannian curvature properties for Cartan spaces. We consider a Cartan space with the mth root metric. We prove that every mth root Cartan space of isotropic Landsberg curvature, or isotropic mean Landsberg curvature, or isotropic mean Berwald curvature reduces to a Landsberg, weakly Landsberg, and weakly Berwald spaces, respectively. Then we show that the mth root Cartan space of almost vanishing H-curvature satisfies H?=?0.     

Gaussian curvature on singular surfaces

We consider prescribing Gaussian curvature on surfaces with conical singularities in both critical and supercritical cases. First we prove a variant of Kazdan-Warner type necessary conditions. Then we obtain sufficient conditions for a function to be the Gaussian curvature of some pointwise conformai singular metric. We only require that the values of the function are not too large at singular points of the metric with the smallest angle, say, less or equal to 0, or less than its average value. To prove the results, we apply some new ideas and techniques. One of them is to estimate the total curvature along a certain minimizing sequence by using the “Distribution of Mass Principle” and the behavior of the critical points at infinity.     

Some estimates of the mean curvature of nonparametric surfaces given over domains inR n

Questions about the behavior of the mean curvature of surfaces given in the form of graph X_n+1 = f(x) over an arbitrary domain Ωin ?ⁿ are considered. It is proved, for example, that if mean curvature H is a continuously monotonically increasing function of coordinates x_n+1 in ?ⁿ⁺¹, then the following assertions are fulfilled: a) if Ω = ?ⁿ, then H = 0, that is, the graph is a minimal surface; b) if ?Ω ≠ ø, then $$\mathop {sup}\limits_{x \in \Omega } |H(f(x))| \cdot dist(x;\partial \Omega ) \leqslant 1$$ is true. Different special cases of Ω are considered, for which exact values of the constant on the right-hand side of (^*) are obtained.     

Integral Menger curvature for surfaces

We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a C1,λ-a priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale R>0 which depends only on an upper bound E for the integral Menger curvature Mp(Σ) and the integrability exponent p, and not on the surface Σ itself; below that scale, each surface with energy smaller than E looks like a nearly flat disc with the amount of bending controlled by the (local) Mp-energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the Hölder exponent λ up to the optimal one, λ=1−(8/p), thus establishing a new geometric ‘Morrey–Sobolev’ imbedding theorem.As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature.     

Deformability of surfaces of positive curvature

For surfaces of positive Gaussian curvature bounded away from zero the following statement is proved: A piece of a given surface containing a preassigned finite set of points and having a Lyapunov boundary can be deformed with an arbitrary given (as large as we like) bending at these points under the condition that the area of the piece is sufficiently small.Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 815–823, May, 1976.I thank V. T. Fomenko for guidance.