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# 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: http://dx.doi.org/10.4310/AJM.2016.v20.n4.a5

#### Authors

#### 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

Published 1 November 2016