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Search: MSC category 53C42 ( Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] )

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1. 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

2. 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

3. 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

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