Metric aspects of conic surfaces

Mijia LAI

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PDF(236 KB)
Front. Math. China ›› 2016, Vol. 11 ›› Issue (5) : 1291-1312. DOI: 10.1007/s11464-016-0556-5
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Metric aspects of conic surfaces

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Abstract

We give a survey on various results regarding the metric aspects of conic surfaces with emphasis on the prescribing curvature problem for conic surfaces.

Keywords

Conic surface / prescribing curvature problem / least-pinched metric

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Mijia LAI. Metric aspects of conic surfaces. Front. Math. China, 2016, 11(5): 1291‒1312 https://doi.org/10.1007/s11464-016-0556-5

References

[1]
Bartolucci D. On the best pinching constant of conformal metrics on S2 with one and two conical singularities. J Geom Anal, 2013, 23(2): 855–877
CrossRef Google scholar
[2]
Bartolucci D, De Marchis F, Malchiodi A. Supercritical conformal metrics on surfaces with conical singularities. Int Math Res Not IMRN, 2011, (24): 5625–5643
[3]
Berger M S. Riemannian structure of prescribed Gaussian curvature for compact 2-manifolds. J Differential Geom, 1971, 5: 325–332
[4]
Berman R. A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kähler-Einstein metrics. Adv Math, 2013, 248: 1254–1297
CrossRef Google scholar
[5]
Bernstein J, Mettler T. Two-dimensional gradient Ricci solitons revisited. Int Math Res Not IMRN, 2015, (1): 78–98
[6]
Bourguignon J, Ezin J. Scalar curvature functions in a class of metrics and conformal transformations. Trans Amer Math Soc, 1987, 301: 723–736
CrossRef Google scholar
[7]
Chen C C, Lin C S. A sharp sup+inf inequality for a nonlinear elliptic equation in ℝ2.Comm Anal Geom, 1998, 6(1): 1–19
CrossRef Google scholar
[8]
Chen Q, Wang W, Wu Y, Xu B. Conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces. Pacific J Math, 2015, 273(1): 75–100
CrossRef Google scholar
[9]
Chen W. A Trudinger inequality on surfaces with conical singularities. Proc Amer Math Soc, 1990, 108(3): 821–832
[10]
Chen W, Li C. Prescribing Gaussian curvatures on surfaces with conical singularities. J Geom Anal, 1991, 1(4): 359–372
CrossRef Google scholar
[11]
Chen W, Li C. Qualitative properties of solutions to some nonlinear elliptic equations in ℝ2.Duke Math J, 1993, 71(2): 427–439
CrossRef Google scholar
[12]
Chen W, Li C. Gaussian curvature on singular surfaces. J Geom Anal, 1993, 3(4): 315–334
CrossRef Google scholar
[13]
Chen W, Li C. What kinds of singular surfaces can admit constant curvature? Duke Math J, 1995, 78(2): 437–451
CrossRef Google scholar
[14]
Donaldson S. Kähler metrics with cone singularities along a divisor. In: Pardalos P M, Rassias T M, eds. Essays in Mathematics and its Applications: In Honor of Stephen Smale’s 80th Birthday. Berlin: Springer, 2012, 49–79
CrossRef Google scholar
[15]
Eremenko A. Metrics of positive curvature with conical singularities on the sphere. Proc Amer Math Soc, 2004, 132(11): 3349–3355
CrossRef Google scholar
[16]
Fang H, Lai M. On curvature pinching of conic 2-spheres. arXiv: 1506.05901
[17]
Kazdan J, Warner F. Curvature functions for compact 2-manifolds. Ann of Math, 1974, 99: 14–47
CrossRef Google scholar
[18]
Luo F, Tian G. Lioville equation and spherical convex polytopes. Proc Amer Math Soc, 1992, 116(4): 1119–1129
CrossRef Google scholar
[19]
Mazzeo R, Rubinstein Y, Sesum N. Ricci flow on surfaces with conic singularities. Anal PDE, 2015, 8(4): 839–882
CrossRef Google scholar
[20]
McOwen R C. Conformal metrics in ℝ2 with prescribed Gaussian curvature and positive total curvature. Indiana Univ Math J, 1985, 34: 97–104
CrossRef Google scholar
[21]
McOwen R C. Point singularities and conformal metrics on Riemann surfaces. Proc Amer Math Soc, 1988, 103: 222–224
CrossRef Google scholar
[22]
Mondello G, Panov D. Spherical metrics with conical singularities on a 2-sphere: angle constraints. arXiv: 1505.01994
[23]
Moser J. A sharp form of an inequality by Neil Turdinger. Indian Univ Math J, 1971, 20: 1077–1092
CrossRef Google scholar
[24]
Moser J. On a nonlinear problem in differential geometry. In: Peixoto M, ed. Dynamical System. New York: Academic Press, 1973, 273–280
CrossRef Google scholar
[25]
Phong D H, Song J, Sturm J, Wang X. Ricci flow on S2 with marked points. arXiv: 1407.1118
[26]
Phong D H, Song J, Sturm J, Wang X. Convergence of the conical Ricci flow on S2 to a soliton. arXiv: 1503.04488
[27]
Picard E. De l’intégration de l’équation Δu= eu sur une surface de Riemann fermée. Crell’s J, 1905, 130: 243–258
[28]
Ramos D. Gradient Ricci solitons on surfaces. Preprint, arXiv: 1304.6391
[29]
Ross J, Thomas R. Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics. J Differential Geom, 2011, 88(1): 109–159
[30]
Thurston W. The Geometry and Topology of Three-Manifolds. Princeton: Princeton University Press, 1978, Chap 13
[31]
Tian G. Kähler-Einstein metrics on algebraic manifolds. In: Demailly J-P, Peternell Th, Tian G, Tyurin A N, eds. Transcendental Methods in Algebraic Geometry: Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Cetraro, Italy, July 4–12, 1994. Lecture Notes in Math, Vol 1646.Berlin: Springer, 1996, 143–185
CrossRef Google scholar
[32]
Troyanov M. Prescribing curvature on compact surfaces with conical singularities. Trans Amer Math Soc, 1991, 324(2): 793–821
CrossRef Google scholar
[33]
Umehara M, Yamada K. Metrics of constant curvature 1 with three conical singularities on the 2-sphere. Illinois J Math, 2000, 44(1): 72–94
[34]
Yin H. Ricci flow on surfaces with conical singularities. J Geom Anal, 2010, 20(4): 970–995
CrossRef Google scholar
[35]
Yin H. Ricci flow on surfaces with conical singularities II. arXiv: 1305.4355

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