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Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

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Abstract

Using a geometric flow, we study the following prescribed scalar curvature plus mean curvature problem: Let \((M,g_0)\) be a smooth compact manifold of dimension \(n\ge 3\) with boundary. Given any smooth functions f in M and h on \(\partial M\), does there exist a conformal metric of \(g_0\) such that its scalar curvature equals f and boundary mean curvature equals h? Assume that f and h are negative and the conformal invariant \(Q(M,\partial M)\) is a negative real number, we prove the global existence and convergence of the so-called prescribed scalar curvature plus mean curvature flows. Via a family of such flows together with some additional variational arguments, we prove the existence and uniqueness of positive minimizers of the associated energy functional and give a confirmative answer to the above problem. The same result also can be obtained by sub–super-solution method and subcritical approximations.

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Notes

  1. The proof here requires \(C^0\)-regularity of \(R_{g(t)}\) and \(h_{g(t)}\) near \(t=0\), which is lack due to Theorem 4.1. However, the same argument works well starting from half of the maximal existence time of the flows instead of \(t=0\).

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Acknowledgements

This work was carried out during the first author’s visit at Rutgers University. He is grateful to Professor Yan Yan Li for the invitation to Rutgers University and fruitful discussions. He also would like to thank mathematics department at Rutgers University for its hospitality and financial support. He is supported by NSFC (No. 11201223), A Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 201417), Program for New Century Excellent Talents in University (NCET-13-0271), the travel grants from AMS Fan fund and Hwa Ying foundation at Nanjing University. The second author is supported by the National Research Foundation of Korea (NRF) Grant (No. 201531021.01) funded by the Korea government (MEST).

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Xuezhang Chen

  2. Department of Mathematics, Sogang University, Seoul, 121-742, Korea

    Pak Tung Ho

  3. Department of Mathematics, Rutgers University, 110 Frenlinghuysen Road, Piscataway, NJ, 08854, USA

    Liming Sun

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  1. Xuezhang Chen
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  2. Pak Tung Ho
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  3. Liming Sun
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Correspondence to Pak Tung Ho.

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Chen, X., Ho, P.T. & Sun, L. Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant. Ann Glob Anal Geom 53, 121–150 (2018). https://doi.org/10.1007/s10455-017-9570-4

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  • DOI: https://doi.org/10.1007/s10455-017-9570-4

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