과제정보
The work was supported by The Science and Technology Project of Guangxi (Guike AD21220114), China Postdoctoral Science Foundation (Grant: No.2021M690773) and Key Laboratory of Mathematical and Statistical Model (Guangxi Normal University), Education Department of Guangxi Zhuang Autonomous Region.
참고문헌
- F. J. Almgren, Jr., Plateau's problem: An invitation to varifold geometry, W. A. Benjamin, Inc., New York, 1966.
- V. I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, 62, American Mathematical Society, Providence, RI, 1998. https://doi.org/10.1090/surv/062
- C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), no. 2, 207-216. https://doi.org/10.1007/BF01425510
- H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and PrekopaLeindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal. 22 (1976), no. 4, 366-389. https://doi.org/10.1016/0022-1236(76)90004-5
- Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, translated from the Russian by A. B. Sosinskii, Grundlehren der mathematischen Wissenschaften, 285, Springer-Verlag, Berlin, 1988. https://doi.org/10.1007/978-3-662-07441-1
- L. A. Caffarelli, Interior W2,p estimates for solutions of the Monge-Ampere equation, Ann. of Math. (2) 131 (1990), no. 1, 135-150. https://doi.org/10.2307/1971510
- L. A. Caffarelli, A localization property of viscosity solutions to the Monge-Ampere equation and their strict convexity, Ann. of Math. (2) 131 (1990), no. 1, 129-134. https://doi.org/10.2307/1971509
- J. S. Case, The weighted σk-curvature of a smooth metric measure space, Pacific J. Math. 299 (2019), no. 2, 339-399. https://doi.org/10.2140/pjm.2019.299.339
- J. S. Case and Y. Wang, Boundary operators associated to the σk-curvature, Adv. Math. 337 (2018), 83-106. https://doi.org/10.1016/j.aim.2018.08.004
- S. Y. Cheng and S. T. Yau, On the regularity of the solution of the n-dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976), no. 5, 495-516. https://doi.org/10.1002/cpa.3160290504
- D. Cordero-Erausquin and B. Klartag, Moment measures, J. Funct. Anal. 268 (2015), no. 12, 3834-3866. https://doi.org/10.1016/j.jfa.2015.04.001
- R. Courant, Dirichlet's principle, conformal mapping, and minimal surfaces, reprint of the 1950 original, Springer-Verlag, New York, 1977.
- Q. Dai, N. S. Trudinger, and X.-J. Wang, The mean curvature measure, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 3, 779-800. https://doi.org/10.4171/JEMS/318
- Q. Dai, X. Wang, and B. Zhou, A potential theory for the k-curvature equation, Adv. Math. 288 (2016), 791-824. https://doi.org/10.1016/j.aim.2015.11.003
- U. Dierkes, S. Hildebrandt, and F. Sauvigny, Minimal surfaces, revised and enlarged second edition, Grundlehren der mathematischen Wissenschaften, 339, Springer, Heidelberg, 2010. https://doi.org/10.1007/978-3-642-11698-8
- M. P. do Carmo, Differential geometry of curves & surfaces, Dover Publications, Inc., Mineola, NY, 2016.
- H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418-491. https://doi.org/10.2307/1993504
- R. J. Gardner and A. Zvavitch, Gaussian Brunn-Minkowski inequalities, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5333-5353. https://doi.org/10.1090/S0002-9947-2010-04891-3
- I. M. Gel'fand, S. G. Gindikin, and M.I. Graev, Integral geometry in affine and projective spaces, J Math. Sci. 18 (1982), no. 1, 39-167. https://doi.org/10.1007/BF01098201
- D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
- E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, 80, Birkhauser Verlag, Basel, 1984. https://doi.org/10.1007/978-1-4684-9486-0
- B. Guan and P. Guan, Convex hypersurfaces of prescribed curvatures, Ann. of Math. (2) 156 (2002), no. 2, 655-673. https://doi.org/10.2307/3597202
- P. Guan, J. Li, and Y. Li, Hypersurfaces of prescribed curvature measure, Duke Math. J. 161 (2012), no. 10, 1927-1942. https://doi.org/10.1215/00127094-1645550
- P. Guan and X.-N. Ma, The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equation, Invent. Math. 151 (2003), no. 3, 553-577. https://doi.org/10.1007/s00222-002-0259-2
- P. Guan, C. Ren, and Z. Wang, Global C2-estimates for convex solutions of curvature equations, Comm. Pure Appl. Math. 68 (2015), no. 8, 1287-1325. https://doi.org/10.1002/cpa.21528
- B. Guan and J. Spruck, Boundary-value problems on 𝕊n for surfaces of constant Gauss curvature, Ann. of Math. (2) 138 (1993), no. 3, 601-624. https://doi.org/10.2307/2946558
- F. R. Harvey and H. B. Lawson, Jr., On boundaries of complex analytic varieties. I, Ann. of Math. (2) 102 (1975), no. 2, 223-290. https://doi.org/10.2307/1971032
- Y. Huang, D. Xi, and Y. Zhao, The Minkowski problem in Gaussian probability space, Adv. Math. 385 (2021), Paper No. 107769, 36 pp. https://doi.org/10.1016/j.aim.2021.107769
- J. Jost, Minimal surfaces and Teichmueller theory, Tsing Hua lectures on geometry & analysis (Hsinchu, 1990-1991), 149-211, Int. Press, Cambridge, MA, 1997.
- H. Lewy, On differential geometry in the large. I. Minkowski's problem, Trans. Amer. Math. Soc. 43 (1938), no. 2, 258-270. https://doi.org/10.2307/1990042
- J. Liu, The Lp-Gaussian Minkowski problem, Calc. Var. Partial Differential Equations 61 (2022), no. 1, Paper No. 28, 23 pp. https://doi.org/10.1007/s00526-021-02141-z
- H. Minkowski, Allgemeine Lehrsatzeuber die convexen Polyeder, Nachr. Ges. Wiss. Gottingen, 198-219, 1897.
- H. Minkowski, Volumen und Oberflache, Math. Ann. 57 (1903), no. 4, 447-495. https://doi.org/10.1007/BF01445180
- C. B. Morrey, Jr., The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807-851. https://doi.org/10.2307/1969401
- L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337-394. https://doi.org/10.1002/cpa.3160060303
- R. Osserman, A Survey of Minimal Surfaces, second edition, Dover Publications, Inc., New York, 1986.
- M. Petrache and T. Riviere, The resolution of the Yang-Mills Plateau problem in supercritical dimensions, Adv. Math. 316 (2017), 469-540. https://doi.org/10.1016/j.aim.2017.06.012
- J. T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Mathematical Notes, 27, Princeton University Press, Princeton, NJ, 1981.
- A. V. Pogorelov, The Minkowski multidimensional problem, translated from the Russian by Vladimir Oliker, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, DC, 1978.
- L. A. Santalo, Integral geometry and geometric probability, second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. https://doi.org/10.1017/CBO9780511617331
- F. Santambrogio, Dealing with moment measures via entropy and optimal transport, J. Funct. Anal. 271 (2016), no. 2, 418-436. https://doi.org/10.1016/j.jfa.2016.04.009
- R. Schneider, Convex bodies: the Brunn-Minkowski theory, second expanded edition, Encyclopedia of Mathematics and its Applications, 151, Cambridge University Press, Cambridge, 2014.
- R. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45-76. http://projecteuclid.org/euclid.cmp/1103904790 103904790
- M. Struwe, Plateau's problem and the calculus of variations, Mathematical Notes, 35, Princeton University Press, Princeton, NJ, 1988.
- A. E. Treibergs and S. W. Wei, Embedded hyperspheres with prescribed mean curvature, J. Differential Geom. 18 (1983), no. 3, 513-521. http://projecteuclid.org/euclid.jdg/1214437786
- N. S. Trudinger and X.-J. Wang, The affine Plateau problem, J. Amer. Math. Soc. 18 (2005), no. 2, 253-289. https://doi.org/10.1090/S0894-0347-05-00475-3
- S. T. Yau, Problem section, in Seminar on Differential Geometry, 669-706, Ann. Of Math. Stud., 102, Princeton Univ. Press, Princeton, NJ, 1982.