Finite Total Curvature Research Articles

On complete noncompact submanifolds with constant mean curvature and finite total curvature in Euclidean spaces

In this paper, we consider a complete noncompact n-submanifold M with parallel mean curvature vector h in an Euclidean space. If M has finite total curvature, we prove that M must be minimal, so that M is an affine n-plane if it is strongly stable. This is a generalization of the result on strongly stable complete hypersurfaces with constant mean curvature in \(\mathbb{R}^{n+1}.\)

The embedded singly periodic Scherk-Costa surfaces

We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's result concerning asymmetric complete minimal submanifolds of Open image in new window with finite total curvature.

The structure of weakly stable minimal hypersurfaces

In this short communication, we announce results from our research on the structure of complete noncompact oriented weakly stable minimal hypersurfaces in a manifold of nonnegative sectional curvature. In particular, a complete oriented weakly stable minimal hypersurface in Rm, m > or = 4, must have only one end; any complete noncompact oriented weakly stable minimal hypersurface has only one end if the complete oriented ambient manifold Nm, m > or = 7, has nonnegative sectional curvature and Ricci curvature bounded below by a positive constant; a complete oriented weakly stable minimal hypersurface in Rm, m > or = 4, with finite total scalar curvature is a hyperplane.

Minimal surfaces of finite total curvature in H×R

We prove that finite total curvature minimal surface of H^2xR are characterize by meromorphic extension of Abresch-Rosenberg quadratic differential and the finite total curvature is a multiple of 2pik with integer k. When the surface is a graph, it is asymptotic to a Scherk's type example.

A balancing condition for weak limits of families of minimal surfaces

We prove a balancing condition for weak limits of families of embedded minimal surfaces of finite total curvature. We use it to prove compactness theorems for certain families of minimal surfaces.

The Cauchy problem for the Liouville equation and Bryant surfaces

We give a construction that connects the Cauchy problem for the 2-dimensional elliptic Liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H 3 , and solve both of them. We construct the unique mean curvature one surface in H 3 that passes through a given curve with a given unit normal along it, and provide diverse applications. In particular, topics such as period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. are treated for these surfaces.

Remarks on a construction of Duistermaat

A construction of Duistermaat's, in which an auxiliary holomorphic curve in the line bundle $\mathcal O(4)$ on $\mathsf{P}_1$ is associated to a minimal surface in $R^3$, is described in a new way and generalised to minimal surfaces in $R^n$. If the minimal surface is complete and has finite total curvature then the auxiliary curve is algebraic.

Noncongruent minimal surfaces with the same symmetries and conformal structure

Concerning complete orientable minimal surfaces with finite total curvature in Euclidean three-space, we show for any positive genus the existence of noncongruent examples having the same symmetry group and conformal type.

An application of Moser iteration to complete minimal submanifolds in a sphere

AbstractWe apply the Moser iteration method to obtain a pointwise bound on the norm of the second fundamental form from a bound on its Ln norm for a complete minimal submanifold in a sphere. As an application we show that a complete minimal submanifold in a sphere with finite total curvature and Ricci curvature bounded away from -∞ must be compact. This complements similar results of Osserman and Oliveira in the case the ambient space is the Euclidean and the hyperbolic space respectively.

A Brunn-Minkowski theory for minimal surfaces

The aim of this paper is to motivate the development of a Brunn-Minkowski theory for minimal surfaces. In 1988, H. Rosenberg and E. Toubiana studied a sum operation for finite total curvature complete minimal surfaces in $\mathbb{R}^{3}$ and noticed that minimal hedgehogs of $\mathbb{R}^{3} $ constitute a real vector space [14]. In 1996, the author noticed that the square root of the area of minimal hedgehogs of $\mathbb{R}^{3}$ that are modelled on the closure of a connected open subset of $\mathbb{S}^{2}$ is a convex function of the support function [5]. In this paper, the author (i) gives new geometric inequalities for minimal surfaces of $\mathbb{R}^{3}$; (ii) studies the relation between support functions and Enneper-Weierstrass representations; (iii) introduces and studies a new type of addition for minimal surfaces; (iv) extends notions and techniques from the classical Brunn-Minkowski theory to minimal surfaces. Two characterizations of the catenoid among minimal hedgehogs are given.

An analogue of the UP-iteration for constant mean curvature one surfaces in hyperbolic 3-space

The UP-iteration is a method for constructing new rational minimal surfaces of genus zero from a given rational minimal surface. Here, an analogue of the UP-iteration is given for constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space H 3 . This yields countably many new complete CMC-1 surfaces of finite total curvature in H 3 , which admit isometric deformations preserving principal curvatures.

On minimal submanifolds in an Euclidean space

AbstractThe concepts “super stable” and “super index” for minimal submanifolds in a Euclidean space are introduced. These concepts coincide with the usual concepts “stable” and “index” when the submanifolds have codimension one. We prove that the only complete super stable minimal submanifolds of finite total scalar curvature and of dimension not less than three in a Euclidean space are affine planes. We also prove that a complete minimal submanifold of dimension larger or equal to three in a Euclidean space with finite super index has finitely many ends. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Some Picard theorems for minimal surfaces

This paper deals with the study of those closed subsets F ⊂ R 3 F \subset \mathbb {R}^3 for which the following statement holds: If S S is a properly immersed minimal surface in R 3 \mathbb {R}^3 of finite topology that is eventually disjoint from F , F, then S S has finite total curvature. The same question is also considered when the conclusion is finite type or parabolicity.

Flux for Bryant surfaces and applications to embedded ends of finite total curvature

We compute the flux of Killing fields through ends of constant mean curvature $1$ in hyperbolic space, and we prove a result conjectured by Rossman, Umehara and Yamada: the flux matrix defined by these authors is equivalent to the flux of Killing fields. We next give a geometric description of embedded ends of finite total curvature. In particular, we show that if such an end is asymptotic to a catenoid cousin, then we can associate an axis to it. We also compute the flux of Killing fields through these ends, and we deduce some geometric properties and analogies to minimal surfaces in Euclidean space.

Total curvature and length estimate for curves in CAT( K) spaces

We introduce the notion of total curvature of curves (which agrees with the usual one in the piecewise smooth case) in spaces of Alexandrov curvature bounded above. Basic properties of total curvature, including rectifiability of curves of finite total curvature and additivity of total curvature, are then obtained. A sharp upper estimate of a type due to Schmidt on the length of a curve in a CAT( K) space is also given in terms of its total curvature and the distance between its endpoints.

Minimal Surfaces Obtained by Ribaucour Transformations

We consider Ribaucour transformations between minimal surfaces and we relate such transformations to generating planar embedded ends. Applying Ribaucour transformations to Enneper's surface and to the catenoid, we obtain new families of complete, minimal surfaces, of genus zero, immersed in R3, with infinitely many embedded planar ends or with any finite number of such ends. Moreover, each surface has one or two nonplanar ends. A particular family is obtained from the catenoid, for each pair (n,m), n≠m, such that nm0 is an irreducible rational number. For any such pair, we get a 1-parameter family of finite total curvature, complete minimal surfaces with n+2 ends, n embedded planar ends and two nonplanar ends of geometric index m, whose total curvature is −4π(n+m). The analytic interpretation of a Ribaucour transformation as a Backlund type transformation and a superposition formula for the nonlinear differential equation ΔΦ= e-2Φ is included.

Existence of algebraic minimal surfaces for an arbitrary puncture set

We will show that any punctured Riemann surface can be conformally immersed into a Euclidean 3-space as a branched complete minimal surface of finite total curvature called an algebraic minimal surface.

Surfaces de Bryant dans [formula omitted] de type fini

We define the notion of Bryant surfaces of finite type: an annular end of a Bryant surface is said to be of finite type if its hyperbolic Gauss map is of finite growth (in the sense of Nevanlinna), and a Bryant surface is said to be of finite type if it is of finite conformal type and if all its ends are of finite type. We prove that a Bryant surface of finite total curvature is of finite type.

PROPERLY EMBEDDED MINIMAL ANNULI BOUNDED BY A CONVEX CURVE

We prove that given a convex Jordan curve , the space of properly embedded minimal annuli in the half-space . Moreover, for a fixed positive number , the exterior Plateau problem that consists of finding a properly embedded minimal annulus in the upper half-space, with finite total curvature, boundary has exactly zero, one or two solutions, each one with a different stability character for the Jacobi operator.AMS 2000 Mathematics subject classification: Primary 53A10. Secondary 49Q05; 53C42

PROPERLY EMBEDDED MINIMAL ANNULI BOUNDED BY A CONVEX CURVE

We prove that given a convex Jordan curve , the space of properly embedded minimal annuli in the half-space . Moreover, for a fixed positive number , the exterior Plateau problem that consists of finding a properly embedded minimal annulus in the upper half-space, with finite total curvature, boundary has exactly zero, one or two solutions, each one with a different stability character for the Jacobi operator.AMS 2000 Mathematics subject classification: Primary 53A10. Secondary 49Q05; 53C42