Search results
Search: MSC category 53C42
( Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] )
1. CJM 2015 (vol 67 pp. 1411)
 Kawakami, Yu

Functiontheoretic Properties for the Gauss Maps of Various Classes of Surfaces
We elucidate the geometric background of functiontheoretic properties
for the Gauss maps of
several classes of immersed surfaces in threedimensional space
forms, for example, minimal surfaces in Euclidean threespace, improper affine spheres in the affine threespace, and constant
mean curvature one surfaces and flat surfaces in hyperbolic threespace. To achieve this purpose, we prove an optimal curvature bound
for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for
the Gauss maps of these classes of surfaces.
Keywords:Gauss map, minimal surface, constant mean curvature surface, front, ramification, omitted value, the Ahlfors island theorem, unicity theorem. Categories:53C42, 30D35, 30F45, 53A10, 53A15 

2. CJM 2009 (vol 61 pp. 641)
 Maeda, Sadahiro; Udagawa, Seiichi

Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions
For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form
$\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we
show that if the mean curvature vector of $M^n$ is parallel and the
sectional curvature $K$ of $M^n$ satisfies some inequality, then
the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is
parallel and our manifold $M^n$ is a space form.
Keywords:space forms, parallel isometric immersions, isotropic immersions, totally umbilic, Veronese manifolds, sectional curvatures, parallel mean curvature vector Categories:53C40, 53C42 

3. CJM 2006 (vol 58 pp. 381)
 Jakobson, Dmitry; Nadirashvili, Nikolai; Polterovich, Iosif

Extremal Metric for the First Eigenvalue on a Klein Bottle
The first eigenvalue of the Laplacian on a surface can be viewed
as a functional on the space of Riemannian metrics of a given
area. Critical points of this functional are called extremal
metrics. The only known extremal metrics are a round sphere, a
standard projective plane, a Clifford torus and an equilateral
torus. We construct an extremal metric on a Klein bottle. It is a
metric of revolution, admitting a minimal isometric embedding into
a sphere ${\mathbb S}^4$ by the first eigenfunctions. Also, this
Klein bottle is a bipolar surface for Lawson's
$\tau_{3,1}$torus. We conjecture that an extremal metric for the
first eigenvalue on a Klein bottle is unique, and hence it
provides a sharp upper bound for $\lambda_1$ on a Klein bottle of
a given area. We present numerical evidence and prove the first
results towards this conjecture.
Keywords:Laplacian, eigenvalue, Klein bottle Categories:58J50, 53C42 

4. CJM 2005 (vol 57 pp. 1291)
 Riveros, Carlos M. C.; Tenenblat, Keti

Dupin Hypersurfaces in $\mathbb R^5$
We study Dupin
hypersurfaces in $\mathbb R^5$ parametrized by lines of curvature, with
four distinct principal curvatures. We characterize locally a generic
family of such hypersurfaces in terms of the principal curvatures and
four vector valued functions of one variable. We show that these vector
valued functions are invariant by inversions and homotheties.
Categories:53B25, 53C42, 35N10, 37K10 
