A flow approach to the prescribed Gaussian curvature problem in \(\mathbb{H}^{\mathrm{n} +1} \). (English) Zbl 1542.35224
Summary: In this paper, we study the following prescribed Gaussian curvature problem:
\[
K=\frac{\tilde{f}(\theta)}{\phi(\rho)^{\alpha-2}\sqrt{\phi(\rho)^2+\vert\overline{\nabla}\rho\vert^2}},
\]
a generalization of the Alexandrov problem (\(\alpha=n+1\)) in hyperbolic space, where \(\tilde{f}\) is a smooth positive function on \(\mathbb{S}^n\), \( \rho\) is the radial function of the hypersurface, \(\phi(\rho)=\sinh\rho\) and \(K\) is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when \(\alpha\geq n+1\). Our argument provides a parabolic proof in smooth category for the Alexandrov problem in \(\mathbb{H}^{n+1}\). We also consider the cases \(2<\alpha\leq n+1\) under the evenness assumption of \(\tilde{f}\) and prove the existence of solutions to the above equations.
MSC:
| 35K55 | Nonlinear parabolic equations |
| 35K93 | Quasilinear parabolic equations with mean curvature operator |
| 53E40 | Higher-order geometric flows |
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