Asian Journal of Mathematics

Volume 20 (2016)

Number 4

On asymptotic plateau’s problem for CMC hypersurfaces on rank 1 symmetric spaces of noncompact type

Pages: 695 – 708

DOI: https://dx.doi.org/10.4310/AJM.2016.v20.n4.a5

Authors

Jean-Baptiste Casteras (Instituto de Matemática, Universidad Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil)

Jaime B. Ripoll (Instituto de Matemática, Universidad Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil)

Abstract

Let $M^n , n \geq 3$, be a Hadamard manifold with strictly negative sectional curvature $K_M \leq -\alpha, \alpha \gt 0$. Assume that $M$ satisfies the strict convexity condition at infinity according to [18] (see also the definition below) and, additionally, that $M$ admits a helicoidal one parameter subgroup $ \{\varphi_t \}$ of isometries (i.e. there exists a geodesic $\gamma$ of $M$ such that $\varphi_t (\gamma (s)) = \gamma (t + s)$ for all $s, t \in \mathbb{R})$. We then prove that, given a compact topological $ \varphi_t \}$−starshaped hypersurface $\Gamma$ in the asymptotic boundary $\partial_{\infty} M$ of $M$ (that is, the orbits of the extended action of $ \{\varphi_t \}$ to $\partial_{\infty} \} M$ intersect $\Gamma$ at one and only one point), and given $H \in \mathbb{R}, \lvert H \rvert \lt \sqrt{\alpha}$, there exists a complete properly embedded constant mean curvature (CMC) $H$ hypersurface $S$ of $M$ such that $\partial_{\infty} S = \Gamma$.

This result extends Theorem 1.8 of B. Guan and J. Spruck [11] to more general ambient spaces, as rank 1 symmetric spaces of noncompact type, and allows $\Gamma$ to be starshaped with respect to more general one parameter subgroup of isometries of the ambient space. For example, in $\mathbb{H}^n , \Gamma$ can be starshaped with respect to a family of loxodromic curves (that includes, in particular, the radial one parameter subgroup of conformal diffeormophisms of $\partial_{\infty} \mathbb{H}^n$ considered in [11]). A fundamental result used to prove our main theorem, which has interest on its own, is the extension of the interior gradient estimates for CMC Killing graphs proved in Theorem 1 of [7] to CMC graphs of Killing submersions.

Keywords

Hadamard manifolds, Killing graphs, asymptotic Dirichlet problem, asymptotic Plateau problem

2010 Mathematics Subject Classification

53C20, 53C21, 58J32

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Published 1 November 2016