Height estimates for surfaces with some constant curvature in $\mathbb{r} \times_{f} \mathbb{r}^{2}$

TL;DR

The paper investigates height bounds and rigidity for surfaces of constant curvature immersed in warped product spaces $\mathbb{R}\times_f\mathbb{M}^2(\kappa)$. It develops conformal-parameter formulations of the immersion equations: a conformal framework for surfaces with positive extrinsic curvature $K_e>0$ and another for mean curvature $H>0$, connecting intrinsic surface data to the ambient warped-product geometry via derivatives of the warping function $f$. The main contributions are height estimates for graph-type surfaces in $\mathbb{R}\times_f\mathbb{R}^2$ under mild monotonicity assumptions on $f$ (yielding $h\le e^{f(0)}/\sqrt{K_e}$ for $K_e>0$ and $h\le e^{f(0)}/H$ for $H>0$) and a classification result for compact minimal graphs, showing that under certain boundary and $f$-monotonicity conditions the only minimal graphs are slices. These results extend prior height-estimate and intrinsic-parameter methods from product spaces to the broader warped-product setting, offering new geometric control for graphs in warped products and generalizing several earlier works (AEG, EGR, FPS, GIR).

Abstract

In this paper, we obtain the necessary equations in a conformal parameter induced by the first or second fundamental forms for a surface that is isometrically immersed in the warped product $\mathbb{R} \times_{f} \mathbb{M}^{2}(κ)$ where $\mathbb{M}^{2}(κ)$ denotes the complete, connected, simply connected, two-dimensional space form of constant curvature. The surface we will consider has either positive extrinsic curvature or positive mean curvature. In each case, we carry out some geometric applications to the theory of constant curvature surfaces immersed in $\mathbb{R} \times_{f} \mathbb{R}^{2}$ under certain conditions on the warping function $f$. Specifically, we derive height estimates for graph-type surfaces with either positive constant extrinsic curvature or positive constant mean curvature. In particular, we classify compact minimal graphs in such warped products. This article extends previous work on the study of constant curvature surfaces immersed in product spaces using conformal parameters, as well as the height estimates for constant curvature surfaces in the warped product $\mathbb{R} \times_{f} \mathbb{R}^{2}$.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.2
• Theorem 2.3: CFP
• Remark 2.4
• Remark 3.1
• Lemma 3.2: AEGAEG2
• proof
• Lemma 3.3