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Masaaki Umehara

Publications and source records attributed to Masaaki Umehara.

82 records · Page 5Linked to original sources
arXiv2001

Period problems for mean curvature one surfaces in H^3 (with application to surfaces of low total curvature)

We survey our recent results on classifying complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute curvature which is the total absolute curvature of the dual CMC-1 surface. Here we discuss results on both notions (proven in two other papers by the authors), and we introduce some new results (with proofs) as well.

math.DG↗
arXiv2001

Minimal surfaces that attain equality in the Chern-Osserman inequality

In the previous paper, Takahasi and the authors generalized the theory of minimal surfaces in Euclidean n-space to that of surfaces with holomorphic Gauss map in certain class of non-compact symmetric spaces. It also includes the theory of constant mean curvature one surfaces in hyperbolic 3-space. Moreover, a Chern-Osserman type inequality for such surfaces was shown. Though its equality condition is not solved yet, the authors have noticed that the equality condition of the original Chern-Osserman inequality itself is not found in any literature except for the case n=3, in spite of its importance. In this paper, a simple geometric condition for minimal surfaces that attains equality in the Chern-Osserman inequality is given. The authors hope it will be a useful reference for readers.

math.DG↗
arXiv2000

Sextactic points on a simple closed curve

We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations.

math.DG↗
arXiv2000

Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature I

A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute curvature which is the total absolute curvature of the dual CMC-1 surface. In this paper, we completely classify CMC-1 surfaces with dual total absolute curvature at most 4π. Moreover, we give new examples and partially classify CMC-1 surfaces with dual total absolute curvature at most 8π.

math.DG↗
arXiv1997

An inverse problem of the flux for minimal surfaces

For a complete minimal surface in the Euclidean 3-space, the so-called flux vector corresponds to each end. The flux vectors are balanced, i.e., the sum of those over all ends are zero. Consider the following inverse problem: For each balanced n vectors, find an n-end catenoid which attains given vectors as flux. Here, an n-end catenoid is a complete minimal surface of genus 0 with ends asymptotic to the catenoids. In this paper, the problem is reduced to solving algebraic equation. Using this reduction, it is shown that, when n=4, the inverse problem for 4-end catenoid has solutions for almost all balanced 4 vectors. Further obstructions for n-end catenoids with parallel flux vectors are also discussed.

dg-ga↗
arXiv1997

General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux

For each end of complete minimal surface in the Euclidean 3-space, the flux vector is defined. It is well-known that the sum of the flux vector over all ends are zero. Consider the following inverse problem: For each balanced n-vectors, find an n-end catenoid which realizes these vectors as flux. Here, an n-end catenoid is a complete minimal surface of genus zero with ends asymptotic to the catenoids. In this paper, we show that the inverse problem can be solved for almost all balanced n vectors for arbitrary n, which is grater than 4. The assumption "almost all" is needed because nonexistence is known for special balanced vectors. We remark that in the case of n=4, the same result has been obtained by the authors (dg-ga/9709006). And the case n=3 is treated by Lopez and Berbanel.

dg-ga↗
arXiv1997

Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus

In this work we give a method for constructing a one-parameter family of complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space that correspond to a given complete minimal surface with finite total curvature in Euclidean 3-space. We show that this one-parameter family of surfaces with the same symmetry properties exists for all given minimal surfaces satisfying certain conditions. The surfaces we construct in this paper are irreducible, and in the process of showing this, we also prove some results about the reducibility of surfaces. Furthermore, in the case that the surfaces are of genus 0, we are able to make some estimates on the range of the parameter for the one-parameter family.

dg-ga↗