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Hypersurfaces of Prescribed Mixed Weingarten Curvature

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Abstract

The primary objectives of this paper can be categorized into two main aspects. First, we investigate the existence of a closed hypersurface in Euclidean space with prescribed mixed quotient type Weingarten curvature. This result can be viewed as an improvement of the result of Chen-Tu-Xiang [12, Theorem 1.1]. Second, we study the homogeneous mixed Weingarten curvature problem and the convexity of the corresponding hypersurface.

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References

  1. Aeppli, A.: On the uniqueness of compact solutions for certain elliptic differential equations. Proc. Am. Math. Soc. 11, 832–836 (1960)

    Article  MathSciNet  Google Scholar 

  2. Alexandrov, A.D.: Uniqueness theorems for surfaces in the large. I (Russian). Vestnik Leningrad. Univ. 11(19), 5–17 (1956)

    MathSciNet  Google Scholar 

  3. Andrews, B.: Pinching estimates and motion of hypersurfaces by curvature functions. J. Reine Angew. Math. 608, 17–33 (2007)

    MathSciNet  Google Scholar 

  4. Andrade, F.J., Barbosa, J.L., de Lira, J.H.: Closed Weingarten hypersurfaces in warped product manifolds. Indiana Univ. Math. J. 58(4), 1691–1718 (2009)

    Article  MathSciNet  Google Scholar 

  5. Bakelman, I., Ja., Kantor, B.E.: Existence of spherically homeomorphic hypersurfaces in Euclidean space with prescribed mean curvature (Russian). Geom. Topol 1, 3–10 (Gos. Ped. Inst. im, Gercena, Leningrad) (1974)

  6. Barbosa, J.L., de Lira, J.H., Oliker, V.I.: A priori estimates for starshaped compact hypersurfaces with prescribed mth curvature function in space forms. In: Nonlinear Problems in Mathematical Physics and Related Topics, vol. I, pp. 35-52. Int. Math. Ser. (N. Y.), 1, Kluwer/Plenum, New York (2002)

  7. Caffarelli, L., Nirenberg, L., Spruck, J.: Nonlinear second order elliptic equations, IV. Starshaped compact Weingarten hypersurfaces. In: Ohya, Y., Kosahara, N., Shimakura, N. (eds.) Current Topics in PDE’s, pp. 1–26. Kinokunia Company LTD, Tokyo (1986)

  8. Caffarelli, L., Guan, P., Ma, X.N.: A constant rank theorem for solutions of fully nonlinear elliptic equations. Commun. Pure Appl. Math. 60, 1769–1791 (2007)

    Article  MathSciNet  Google Scholar 

  9. Chen, C., Dong, W., Han, F.: Interior Hessian estimates for a class of Hessian type equations. Calc. Var. Partial Differ. Equ. 62, 52 (2023)

    Article  MathSciNet  Google Scholar 

  10. Chen, C., Xu, L.: The \(L_p\) Minkowski type problem for a class of mixed Hessian quotient equations. Adv. Math. 411, 108794 (2022)

    Article  Google Scholar 

  11. Chen, D., Li, H., Wang, Z.: Starshaped compact hypersurfaces with prescribed Weingarten curvature in warped product manifolds. Calc. Var. Partial Diff. Equ. 57, 42 (2018)

    Article  MathSciNet  Google Scholar 

  12. Chen, X., Tu, Q., Xiang, N.: A class of Hessian quotient equations in Euclidean space. J. Differ. Equ. 269, 11172–11194 (2020)

    Article  MathSciNet  Google Scholar 

  13. Cheng, S.Y., Yau, S.T.: On the regularity of the solution of the \(n\)-dimensional Minkowski problem. Commun. Pure Appl. Math. 29, 495–516 (1976)

    Article  MathSciNet  Google Scholar 

  14. Chern, S.S.: Integral formulas for hypersurfaces in Euclidean space and their applications to uniqueness theorems. J. Math. Mech. 8, 947–955 (1959)

    MathSciNet  Google Scholar 

  15. Chu, J., Jiao, H.: Curvature estimates for a class of Hessian type equations. Calc. Var. Partial Differ. Equ. 60(3), 1–18 (2021)

    Article  MathSciNet  Google Scholar 

  16. Fu, J., Wang, Z., Wu, D.: Form-type Calabi-Yau equations. Math. Res. Lett. 17(5), 887–903 (2010)

    Article  MathSciNet  Google Scholar 

  17. Guan, B., Guan, P.: Convex hypersurfaces of prescribed curvatures. Ann. Math. 156(2), 655–673 (2002)

    Article  MathSciNet  Google Scholar 

  18. Guan, P., Ma, X.: The Christoffel-Minkowski problem I: convexity of solutions of a Hessian equation. Invent. Math. 151(3), 553–577 (2003)

    Article  MathSciNet  Google Scholar 

  19. Guan, P., Lin, C., Ma, X.: The Christoffel-Minkowski problem II: Weingarten curvature equations. Chin. Ann. Math. Ser. B. 27(6), 595–614 (2006)

    Article  MathSciNet  Google Scholar 

  20. Guan, P., Ma, X., Zhou, F.: The Christofel-Minkowski problem III: existence and convexity of admissible solutions. Commun. Pure Appl. Math. 59, 1352–1376 (2006)

    Article  MathSciNet  Google Scholar 

  21. Han, F., Ma, X., Wu, D.: The existence of \(k\)-convex hypersurface with prescribed mean curvature. Calc. Var. Partial Differ. Equ. 42(1), 43–72 (2011)

    Article  MathSciNet  Google Scholar 

  22. Harvey, F.R., Lawson, H.B., Jr.: Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds. J. Differ. Geom. 88(3), 395–482 (2011)

    Article  MathSciNet  Google Scholar 

  23. Harvey, F.R., Lawson, H.B., Jr.: Geometric plurisubharmonicity and convexity: an introduction. Adv. Math. 230(4–6), 2428–2456 (2012)

    Article  MathSciNet  Google Scholar 

  24. Harvey, F.R., Lawson, H.B., Jr.: Existence, uniqueness and removable singularities for nonlinear partial differential equations in geometry. Surv. Differ. Geom. 18(1), 103–156 (2013)

    Article  MathSciNet  Google Scholar 

  25. Jin, Q., Li, Y.: Starshaped compact hypersurfaces with prescribed k-th mean curvature in hyperbolic space. Discrete Contin. Dyn. Syst. 15(2), 367–377 (2006)

    Article  MathSciNet  Google Scholar 

  26. Lewy, H.: On differential geometry in the large, I (Minkowski’s problem). Trans. Am. Math. Soc. 43, 258–270 (1938)

    MathSciNet  Google Scholar 

  27. Lieberman, G.: Second Order Parabolic Differential Equations. World Scientific, Singapore (1996)

    Book  Google Scholar 

  28. Li, Y., Oliker, V.I.: Starshaped compact hypersurfaces with prescribed m-th mean curvature in elliptic space. J. Partial Diff. Equ. 15(3), 68–80 (2002)

    MathSciNet  Google Scholar 

  29. Minkowski, H.: Allgemeine Lehrsätze über die konvexen Polyeder. Nachr. Ges. Wiss. Gottingen 198–219 (1897)

  30. Nirenberg, L.: The Weyl and Minkowski problems in differential geometry in the large. Commun. Pure Appl. Math. 6, 337–394 (1953)

    Article  MathSciNet  Google Scholar 

  31. Pogorelov, A.: Regularity of a convex surface with given Gaussian curvature. (Russian) Mat. Sbornik N.S. 31(73), 88–103 (1952)

  32. Reilly, R.C.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differ. Geom. 8(3), 465–477 (1973)

    Article  MathSciNet  Google Scholar 

  33. Sha, J.: p-convex Riemannian manifolds. Invent. Math. 83(3), 437–447 (1986)

    Article  MathSciNet  Google Scholar 

  34. Sha, J.: Handlebodies and p-convexity. J. Differ. Geom. 25(3), 353–361 (1987)

    Article  MathSciNet  Google Scholar 

  35. Székelyhidi, G., Tosatti, V., Weinkove, B.: Gauduchon metrics with prescribed volume form. Acta Math. 219(1), 181–211 (2017)

    Article  MathSciNet  Google Scholar 

  36. Tosatti, V., Weinkove, B.: The Monge-Ampère equation for \((n-1)\)-plurisubharmonic functions on a compact Kähler manifold. J. Am. Math. Soc. 30, 311–346 (2017)

    Article  Google Scholar 

  37. Treibergs, A.E., Wei, S.W.: Embedded hypersurfaces with prescribed mean curvature. J. Differ Geom. 18(3), 513–521 (1983)

    Article  Google Scholar 

  38. Wu, H.: Manifolds of partially positive curvature. Indiana Univ. Math. J. 36(3), 525–548 (1987)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Xinqun Mei & Hua Zhu

  2. Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, 79104, Freiburg im Breisgau, Germany

    Xinqun Mei

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  1. Xinqun Mei
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  2. Hua Zhu
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Correspondence to Hua Zhu.

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Mei, X., Zhu, H. Hypersurfaces of Prescribed Mixed Weingarten Curvature. J Geom Anal 34, 64 (2024). https://doi.org/10.1007/s12220-023-01515-3

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