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\({\mathcal {M}}\)-Convex Hypersurfaces with Prescribed Shifted Gaussian Curvature in Warped Product Manifolds

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Abstract

In this paper, we first give a notion of \({\mathcal {M}}\)-convexity, and then under suitable settings related to this superconvexity, we can obtain the existence of solutions to prescribed shifted Gaussian curvature equations in warped product manifolds of special type by the standard degree theory based on the a priori estimates for the solutions. This is to say that the existence of \({\mathcal {M}}\)-convex, closed hypersurface (which is graphic with respect to the base manifold and whose shifted Gaussian curvature satisfies some constraint) in a given warped product manifold of special type can be assured. Besides, different from prescribed Weingarten curvature problems in space forms, due to the \({\mathcal {M}}\)-convexity of hypersurfaces in the warped product manifold considered, we do not need to impose a sign condition for the radial derivative of the prescribed function in the shifted Gaussian curvature equation to prove the existence of solutions.

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Notes

  1. Actually, they are: (1) semidefiniteness of the second fundamental form; (2) nonnegative Ricci curvature; (3) nonnegative sectional curvature; (4) infinitesimal support by horospheres.

  2. Here, H denotes the mean curvature of the hypersurface \(N^{n}\).

  3. Obviously, \({\det }^{\frac{1}{n}}\left( {\bar{h}}_{ij}(X)\right) \) denotes the n-th root of the determinant \(\det \left( {\bar{h}}_{ij}(X)\right) \). In the sequel, the notation \({\det }^{\frac{1}{n}}\left( \cdot \right) \) will be used similarly.

  4. Clearly, for accuracy, here \(D_{i}u\) should be \(D_{e_{i}}u\). In the sequel, without confusion and if needed, we prefer to simplify covariant derivatives like this. In this setting, \(u_{ij}:=D_{j}D_{i}u\), \(u_{ijk}:=D_{k}D_{j}D_{i}u\) mean \(u_{ij}=D_{e_{j}}D_{e_{i}}u\) and \(u_{ijk}=D_{e_{k}}D_{e_{j}}D_{e_{i}}u\), respectively. We will also simplify covariant derivatives on \({\mathcal {G}}\) and \({\bar{M}}\) similarly if necessary.

  5. In this setting, repeated Latin letters should be made summation from 1 to n.

  6.  As explained in a footnote of [20], in \(\mathbb {R}^{n+1}\) or the hyperbolic \((n+1)\)-space \(\mathbb {H}^{n+1}\), there is no need to define the vector field V since these two spaces are two-points homogeneous and global coordinate system can be set up, and then X(x) can be seen as the position vector directly.

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Acknowledgements

This work is partially supported by the NSF of China (Grant Nos. 11801496 and 11926352), the Fok Ying-Tung Education Foundation (China) and Hubei Key Laboratory of Applied Mathematics (Hubei University). The authors would like to thank the anonymous referees for their careful reading and valuable comments such that the paper appears as its present version.

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  1. Faculty of Mathematics and Statistics, Key Laboratory of Applied Mathematics of Hubei Province, Hubei University, Wuhan, 430062, China

    Ya Gao, Jing Mao & Shiyun Sun

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  1. Ya Gao
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Correspondence to Jing Mao.

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Gao, Y., Mao, J. & Sun, S. \({\mathcal {M}}\)-Convex Hypersurfaces with Prescribed Shifted Gaussian Curvature in Warped Product Manifolds. Results Math 79, 23 (2024). https://doi.org/10.1007/s00025-023-02050-x

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