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Isometric immersions of the hyperbolic space $H^n(-1)$ into $H^{n+1}(-1)$

100%

Hu Z.-J.

Colloquium Mathematicum

1999

tom 79

nr 1

17-23

We transform the problem of determining isometric immersions from $H^n(-1)$ into $H^{n+1}(-1)$ into that of solving equations of degenerate Monge-Ampère type on the unit ball $B^n(1)$. By presenting one family of special solutions to the equations, we obtain a great many noncongruent examples of such isometric immersions with or without umbilic set.

On the nonexistence of stable minimal submanifolds and the Lawson-Simons conjecture

63%

Hu Z.-J., Wei G.-X.

Colloquium Mathematicum

2003

tom 96

nr 2

213-223

Let M̅ be a compact Riemannian manifold with sectional curvature $K_{M̅}$ satisfying $1/5 < K_{M̅} ≤ 1$ (resp. $2 ≤ K_{M̅} < 10$), which can be isometrically immersed as a hypersurface in the Euclidean space (resp. the unit Euclidean sphere). Then there exist no stable compact minimal submanifolds in M̅. This extends Shen and Xu's result for 1/4-pinched Riemannian manifolds and also suggests a modified version of the well-known Lawson-Simons conjecture.

Ograniczanie wyników

2 Colloquium Mathematicum

2 Hu Z.-J.

1 Wei G.-X.

1 2003

1 1999

Wyniki wyszukiwania

Isometric immersions of the hyperbolic space $H^n(-1)$ into $H^{n+1}(-1)$

On the nonexistence of stable minimal submanifolds and the Lawson-Simons conjecture