Existence result for minimal hypersurfaces¶with a prescribed finite number of planar ends

Paralleling what has been done for minimal surfaces in ℝ³, we develop a gluing procedure to produce, for any k≥ 2 and any n≥ 3 complete immersed minimal hypersurfaces of ℝ ^{n +1} which have k planar ends. These surfaces are of the topological type of a sphere with k punctures and they all have finite total curvature. Received: 1 July 1999 / Revised version: 31 May 2000     

Existence, uniqueness and graph representation of weighted minimal hypersurfaces

By means of a weight matrix, we introduce the class of weighted minimal hypersurfaces which yield a natural generalisation of minimal surfaces. Generalising a classical result of Radó, we prove existence, uniqueness and graph representation for weighted minimal hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} with prescribed boundary manifold.     

Hypersurfaces whose mean curvature function is of finite type

We define finite mean type hypersurfaces to be hypersurfaces with mean curvature function of finite Chen-type. Then, we prove that hyperplanes are the only polynomial translation hypersurfaces of finite mean type in a Euclidean spaceE ⁿ⁺¹. And we show that the only non-conic hyperquadrics of finite mean type in Euclidean spaces are the hyperspheres and the cylinders on spheres. Finally, we state that, among all hypercylinders in a Euclidean spaceE ⁿ⁺¹, the only ones of finite mean type are those on finite mean type planar curves.     

Affine minimal hypersurfaces of rotation

We give a complete list of affine minimal surfaces inA ³ with Euclidean rotational symmetry, completing the treatise given in [1] and prove that these surfaces have maximal affine surface area within the class of all affine surfaces of rotation satisfying suitable boundary conditions. Besides we show that for rotationally symmetric locally strongly convex affine minimal hypersurfaces inA ⁿ ,n4, the second variation of the affine surface area is negative definite under certain conditions on the meridian.     

Foliations of knot complements in the bicylinder boundary

In this expository paper, we shall analyze a particularly important class of examples of surfaces and hypersurfaces in Euclidean 4-space, namely those which arise by considering real 4-space as the space of twocomplex variablesz andw and by taking geometric loci of the formf(z,w)=0 or hypersurfaces associated with such loci. Such surfaces and hypersurfaces are important in the study of the singularities of algebraic curves, as described for example in the book of Milnor [3], and they have been used recently in the construction of foliations of the 3-dimensional sphere by Lawson [2]. The examples of this paper were first presented at the International Symposium of Dynamical Systems and Foliations at Salvador in the summer of 1971, and the author expresses his gratitude for the opportunity to participate in that conference. The examples constructed in this paper are closely related to another paper of the author [1] concerning minimal surfaces in the bicylinder boundary.     

New extension phenomena for solutions of tangential Cauchy–Riemann equations

In our recent work, we showed that C^∞ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in ?² are not analytic in general. This result raised again the question on the nature of CR-maps of a real-analytic hypersurfaces.

In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-diffeomorphisms which we call the sectorial analyticity property. It appears to be the optimal regularity property for CR-diffeomorphisms in general. We emphasize that such type of extension never appeared previously in the literature. Second, we introduce the class of Fuchsian type hypersurfaces and prove that (infinitesimal generators of) CR-automorphisms of a Fuchsian type hypersurface are still analytic. In particular, this solves a problem formulated earlier by Shafikov and the first author.

Finally, we prove a regularity result for formal CR-automorphisms of Fuchsian type hypersurfaces.     

On the smoothness of the equizonal ovaloids

We show that the n-dimensional equizonal ovaloids are analytic when n is even and are of exactly C ^n-1 smoothness when n is odd. This substantially improves the previously published result on the smoothness of the even-dimensional equizonal ovaloids and slightly corrects the previously published statement regarding the smoothness of the odd-dimensional equizonal ovaloids. Our methods should be generally useful in determining the degree of smoothness of surfaces and hypersurfaces of revolution generated by piecewise-defined profile curves. In particular, they include a novel and elegant application of Bernstein’s theory of absolutely monotonic functions.     

Integer points on curves and surfaces

Various upper bounds are given for the number of integer points on plane curves, on surfaces and hypersurfaces. We begin with a certain class of convex curves, we treat rather general surfaces in ³ which include algebraic surfaces with the exception of cylinders, and we go on to hypersurfaces in ⁿ with nonvanishing Gaussian curvature.Written with partial supports from NSF grant No. MCS-8211461.     

Bernstein type result for constant mean curvature hypersurface

We prove a Bernstein type theorem for constant mean curvature hypersurfaces in ℝ ⁿ⁺¹ under certain growth conditions for n ⩽ 3. Our result extends the case when M is a minimal hypersurface in the same condition.      

Geodesics in minimal surfaces

Abstract—In this paper, we consider connected minimal surfaces in R³ with isothermal coordinates and with a family of geodesic coordinates curves, these surfaces will be called GICM-surfaces. We give a classification of the GICM-surfaces. This class of minimal surfaces includes the catenoid, the helicoid and Enneper’s surface. Also, we show that one family of this class of minimal surfaces has at least one closed geodesic and one 1-periodic family of this class has finite total curvature. As application we show other characterization of catenoid and helicoid. Finally, we show that the class of GICM-surfaces coincides with the class of minimal surfaces whose the geodesic curvature k _g ¹ and k _g ² of the coordinates curves satisfy αk _g ¹ + βk _g ² = 0, α, β ∈ R.     

The Minding formula and its applications

We apply the Minding Formula for geodesic curvature and the Gauss-Bonnet Formula to calculate the total Gaussian curvature of certain 2-dimensional open complete branched Riemannian manifolds, the M\cal M surfaces. We prove that for an M\cal M surface, the total curvature depends only on its Euler characteristic and the local behaviour of its metric at ends and branch points. Then we check that many important surfaces, such as complete minimal surfaces in \Bbb Rⁿ{\Bbb R}^n with finite total curvature, complete constant mean curvature surfaces in hyperbolic 3-space H³ (–1) with finite total curvature, are actually branch point free M\cal M surfaces. Therefore as corollaries we give simple proofs of some classical theorems such as the Chern-Osserman theorem for complete minimal surfaces in \Bbb Rⁿ{\Bbb R}^n with finite total curvature. For the reader's convenience, we also derive the Minding Formula.     

Interpolation and sampling hypersurfaces for the Bargmann-Fock space in higher dimensions

We study those smooth complex hypersurfaces W in having the property that all holomorphic functions of finite weighted L^p norm on W extend to entire functions with finite weighted L^p norm. Such hypersurfaces are called interpolation hypersurfaces. We also examine the dual problem of finding all sampling hypersurfaces, i.e., smooth hypersurfaces W in such that any entire function with finite weighted L^p norm is stably determined by its restriction to W. We provide sufficient geometric conditions on the hypersurface to be an interpolation or sampling hypersurface. The geometric conditions that imply the extension property and the restriction property are given in terms of some directional densities. The first author is supported by projects MTM 2005-08984-Co2-O2 and 2001SGR00611 The third author is partially supported by NSF grant DMS0400909     

Geometry of Holomorphic Distributions of Real Hypersurfaces in a Complex Projective Space

We characterize homogeneous real hypersurfaces M's of type (A ₁), (A ₂) and (B) of a complex projective space in the class of real hypersurfaces by studying the holomorphic distribution T ⁰ M of M.     

Cyclic polytopes and d-order curves

A cyclic d-polytope is a convex polytope combinatorially equivalent to the convex hull of a finite subset of a d-order curve in R ^d. We give an affirmative answer to a conjecture of M. A. Perles [4] by proving that every even-dimensional cyclic polytope occurs in this way: its set of vertices can always be extended to a d-order curve.     

Biharmonic hypersurfaces in Euclidean space E5

In this paper, we study biharmonic hypersurfaces in E⁵. We prove that every biharmonic hypersurface in Euclidean space E⁵ must be minimal.     

On a projectively minimal hypersurface in the unimodular affine space

We derive a differential equation defining a projectively minimal hypersurface in the affine space A ⁿ⁺¹. Several examples of such hypersurfaces are given. We prove that only an ellipsoid is a projectively minimal surface in A ³ which is compact and strongly convex.     

Cyclic surfaces in E 5 generated by equiform motions

In this paper, we study cyclic surfaces in E⁵ generated by equiform motions of a circle. The properties of this cyclic surfaces up to the first order are discussed. We prove the following new result: A cyclic 2-surfaces in E⁵ in general are contained in canal hypersurfaces. Finally we give an example.     

Index and topology of minimal hypersurfaces in $$\mathbb {R}^n$$

In this paper, we consider immersed two-sided minimal hypersurfaces in \(\mathbb {R}^n\) with finite total curvature. We prove that the sum of the Morse index and the nullity of the Jacobi operator is bounded from below by a linear function of the number of ends and the first Betti number of the hypersurface. When \(n=4\), we are able to drop the nullity term by a careful study for the rigidity case. Our result is the first effective Morse index bound by purely topological invariants, and is a generalization of Li and Wang (Math Res Lett 9(1):95–104, 2002). Using our index estimates and ideas from the recent work of Chodosh–Ketover–Maximo (Minimal surfaces with bounded index, 2015. arXiv:1509.06724), we prove compactness and finiteness results of minimal hypersurfaces in \(\mathbb {R}^4\) with finite index.