Minimal surfaces in $$S^3$$ : a survey of recent results
Tóm tắt
In this survey, we discuss various aspects of the minimal surface equation in the three-sphere
$$S^3$$
. After recalling the basic definitions, we describe a family of immersed minimal tori with rotational symmetry. We then review the known examples of embedded minimal surfaces in
$$S^3$$
. Besides the equator and the Clifford torus, these include the Lawson and Kapouleas-Yang examples, as well as a new family of examples found recently by Choe and Soret. We next discuss uniqueness theorems for minimal surfaces in
$$S^3$$
, such as the work of Almgren on the genus
$$0$$
case, and our recent solution of Lawson’s conjecture for embedded minimal surfaces of genus
$$1$$
. More generally, we show that any minimal surface of genus
$$1$$
which is Alexandrov immersed must be rotationally symmetric. We also discuss Urbano’s estimate for the Morse index of an embedded minimal surface and give an outline of the recent proof of the Willmore conjecture by Marques and Neves. Finally, we describe estimates for the first eigenvalue of the Laplacian on a minimal surface.
Tài liệu tham khảo
Alexandrov, A.D.: Uniqueness theorems for surfaces in the large I. Vesnik Leningrad Univ. 11, 5–17 (1956)
Almgren, F.J. Jr.: Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem. Ann. Math. 84, 277–292 (1966)
Andrews, B.: Non-collapsing in mean-convex mean curvature flow, arxiv:1108.0247
Andrews, B., Li, H.: Embedded constant mean curvature tori in the three-sphere, arxiv:1204.5007
Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. 36(9), 235–249 (1957)
Bobenko, A.I.: All constant mean curvature tori in \({\mathbb{R}}^3, S^3, {\mathbb{H}}^3\) in terms of theta-functions. Math. Ann. 290, 209–245 (1991)
Bony, J.M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble) 19, 277–304 (1969)
Brendle, S.: Constant mean curvature surfaces in warped product manifolds. Publ. Math. IHÉS (2012). doi:10.1007/s10240-012-0047-5
Brendle, S.: Embedded minimal tori in \(S^3\) and the Lawson conjecture. Acta Math. (to appear)
Brendle, S.: Alexandrov immersed minimal tori in \(S^3\), arxiv:1211.5112
Brendle, S., Eichmair, M.: Isoperimetric and Weingarten surfaces in the Schwarzschild manifold. J. Differ. Geom. (to appear)
Choe, J.: Minimal surfaces in \(S^3\) and Yau’s conjecture. In: Proceedings of the Tenth International Workshop on Differential Geometry, pp. 183–188. Kyungpook National University, Taegu (2006)
Choe, J., Soret, M.: First eigenvalue of symmetric minimal surfaces in \(S^3\). Indiana Univ. Math. J. 58, 269–281 (2009)
Choe, J., Soret, M.: New minimal surfaces in \(S^3\) desingularizing the Clifford tori, available at http://newton.kias.re.kr/~choe/clifford.pdf
Choi, H.I., Schoen, R.: The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature. Invent. Math. 81, 387–394 (1985)
Choi, H.I., Wang, A.N.: A first eigenvalue estimate for minimal hypersurfaces. J. Differ. Geom. 18, 559–562 (1983)
Christodoulou, D., Yau, S.T.: Some remarks on the quasi-local mass. In: Mathematics and General Relativity (Santa Cruz, 1986), Contemporary Mathematics, vol. 71, pp. 9–14. Am. Math. Soc., Providence RI (1986)
Courant, R., Hilbert, D.: Methods of Mathematical Physics, I. Wiley Interscience, New York (1953)
Eichmair, M., Metzger, J.: Large isoperimetric surfaces in initial data sets. J. Differ. Geom. 94, 159–186 (2013)
Eichmair, M., Metzger, J.: On large volume preserving stable CMC surfaces in initial data sets. J. Differ. Geom 91, 81–102 (2012)
Grayson, M.: Shortening embedded curves. Ann. Math. 129, 71–111 (1989)
Hamilton, R.: An isoperimetric estimate for the Ricci flow on the two-sphere, Modern Methods in Complex Analysis (Princeton 1992), 191–200, Ann. Math. Stud. 137, Princeton University Press, Princeton, NJ (1995)
Hitchin, N.S.: Harmonic maps from a \(2\)-torus to the \(3\)-sphere. J. Differ. Geom. 31, 627–710 (1990)
Hsiang, W.Y., Lawson, H.B. Jr.: Minimal submanifolds of low cohomogeneity. J. Differ. Geom. 5, 1–38 (1971)
Huisken, G.: A distance comparison principle for evolving curves. Asian J. Math. 2, 127–133 (1998)
Huisken, G., Yau, S.T.: Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. 124, 281–311 (1996)
Ilmanen, T., and White, B.: Sharp lower bounds on density of area-minimizing cones, arxiv:1010.5068
Kapouleas, N.: Doubling and desingularization constructions for minimal surfaces, Surveys in Geometric Analysis and Relativity, Advanced Lectures in Math, vol. 20, pp. 281–325. International Press, Somerville, MA (2011)
Kapouleas, N., Wiygul, D.: Minimal surfaces in the round three-sphere by desingularizing intersecting Clifford tori (2013, preprint)
Kapouleas, N., Yang, S.D.: Minimal surfaces in the three-sphere by doubling the Clifford torus. Am. J. Math. 132, 257–295 (2010)
Karcher, H., Pinkall, U., Sterling, I.: New minimal surfaces in \(S^3\). J. Differ. Geom. 28, 169–185 (1988)
Korevaar, N., Kusner, R., Ratzkin, J.: On the nondegeneracy of constant mean curvature surfaces. Geom. Funct. Anal. 16, 891–923 (2006)
Korevaar, N., Kusner, R., Solomon, B.: The structure of complete embedded surfaces with constant mean curvature. J. Differ. Geom. 30, 465–503 (1989)
Kusner, R., Mazzeo, R., Pollack, D.: The moduli space of complete embedded constant mean curvature surfaces. Geom. Funct. Anal. 6, 120–137 (1996)
Kuwert, E., Li, Y., Schätzle, R.: The large genus limit of the infimum of the Willmore energy. Am. J. Math. 132, 37–51 (2010)
Lawson Jr, H.B.: Local rigidity theorems for minimal hypersurfaces. Ann. Math. 89, 187–197 (1969)
Lawson Jr, H.B.: Complete minimal surfaces in \(S^3\). Ann. Math. 92, 335–374 (1970)
Lawson Jr, H.B.: The unknottedness of minimal embeddings. Invent. Math. 11, 183–187 (1970)
Lawson Jr, H.B.: Lectures on Minimal Submanifolds. Publish or Perish, Berkeley (1980)
Li, P., Yau, S.T.: A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math. 69, 269–291 (1982)
Marques, F.C., Neves, A.: Min-max theory and the Willmore conjecture, arxiv:1202.6036
Meeks, W., Yau, S.T.: The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z. 179, 151–168 (1982)
Qing, J., Tian, G.: On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat \(3\)-manifolds. J. Am. Math. Soc. 20, 1091–1110 (2007)
Ritoré, M., Ros, A.: Stable constant mean curvature tori and the isoperimetric problem in three space forms. Comment. Math. Helv. 67, 293–305 (1992)
Ros, A.: A two-piece property for compact minimal surfaces in a three-sphere. Indiana Univ. Math. J. 44, 841–849 (1995)
Ros, A.: The Willmore conjecture in the real projective space. Math. Res. Lett. 6, 487–493 (1999)
Ros, A.: The isoperimetric problem, Global theory of minimal surfaces, Clay Math. Proc. vol. 2, Am. Math. Soc., Providence, RI, 175–209 (2005)
Sheng, W., Wang, X.J.: Singularity profile in the mean curvature flow. Methods Appl. Anal. 16, 139–155 (2009)
Simons, J.: Minimal varities in Riemannian manifolds. Ann. Math. 88, 62–105 (1968)
Tang, Z.Z., Yan, W.: Isoparametric foliation and Yau conjecture on the first eigenvalue. J. Differ. Geom. (to appear)
Topping, P.: Towards the Willmore conjecture. Calc. Var. PDE 11, 361–393 (2000)
Urbano, F.: Minimal surfaces with low index in the three-dimensional sphere. Proc. Am. Math. Soc. 108, 989–992 (1990)
Yau, S.T.: Problem section, Seminar on Differential Geometry, Annals of Mathematics Studies, vol. 102, pp. 669–706. Princeton University Press, Princeton (1982)
White, B.: The size of the singular set in mean curvature flow of mean convex sets. J. Am. Math. Soc. 13, 665–695 (2000)
White, B.: The nature of singularities in mean curvature flow of mean convex sets. J. Am. Math. Soc. 16, 123–138 (2003)
White, B.: Subsequent singularities in mean-convex mean curvature flow, arxiv:1103.1469
Willmore, T.J.: Note on embedded surfaces. An. Sti. Univ. ”Al. I. Cuza” Iasi Sect. I a Mat. (N.S.) 11B, 493–496 (1965)
Willmore, T.J.: Mean curvature of Riemannian immersions. J. Lond. Math. Soc. 3, 307–310 (1971)