Metric aspects of conic surfaces

Mijia LAI

Front. Math. China ›› 2016, Vol. 11 ›› Issue (5) : 1291 -1312.

PDF (236KB)
Front. Math. China ›› 2016, Vol. 11 ›› Issue (5) : 1291 -1312. DOI: 10.1007/s11464-016-0556-5
SURVEY ARTICLE
SURVEY ARTICLE

Metric aspects of conic surfaces

Author information +
History +
PDF (236KB)

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

Cite this article

Download citation ▾
Mijia LAI. Metric aspects of conic surfaces. Front. Math. China, 2016, 11(5): 1291-1312 DOI:10.1007/s11464-016-0556-5

登录浏览全文

4963

注册一个新账户 忘记密码

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

[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

[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

[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

[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

[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

[11]

Chen W, Li C. Qualitative properties of solutions to some nonlinear elliptic equations in ℝ2.Duke Math J, 1993, 71(2): 427–439

[12]

Chen W, Li C. Gaussian curvature on singular surfaces. J Geom Anal, 1993, 3(4): 315–334

[13]

Chen W, Li C. What kinds of singular surfaces can admit constant curvature? Duke Math J, 1995, 78(2): 437–451

[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

[15]

Eremenko A. Metrics of positive curvature with conical singularities on the sphere. Proc Amer Math Soc, 2004, 132(11): 3349–3355

[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

[18]

Luo F, Tian G. Lioville equation and spherical convex polytopes. Proc Amer Math Soc, 1992, 116(4): 1119–1129

[19]

Mazzeo R, Rubinstein Y, Sesum N. Ricci flow on surfaces with conic singularities. Anal PDE, 2015, 8(4): 839–882

[20]

McOwen R C. Conformal metrics in ℝ2 with prescribed Gaussian curvature and positive total curvature. Indiana Univ Math J, 1985, 34: 97–104

[21]

McOwen R C. Point singularities and conformal metrics on Riemann surfaces. Proc Amer Math Soc, 1988, 103: 222–224

[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

[24]

Moser J. On a nonlinear problem in differential geometry. In: Peixoto M, ed. Dynamical System. New York: Academic Press, 1973, 273–280

[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

[32]

Troyanov M. Prescribing curvature on compact surfaces with conical singularities. Trans Amer Math Soc, 1991, 324(2): 793–821

[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

[35]

Yin H. Ricci flow on surfaces with conical singularities II. arXiv: 1305.4355

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg

AI Summary AI Mindmap
PDF (236KB)

914

Accesses

0

Citation

Detail

Sections
Recommended

AI思维导图

/