Surfaces of bounded mean curvature in Riemannian manifolds

Gadgil, Siddhartha; Seshadri, Harish

doi:10.1090/S0002-9947-2011-05190-1

Surfaces of bounded mean curvature in Riemannian manifolds
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by Siddhartha Gadgil and Harish Seshadri PDF

Trans. Amer. Math. Soc. 363 (2011), 3977-4005 Request permission

Abstract:

Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on the norm of mean curvature and area. We show that on passing to a subsequence, we can choose parametrisations of the surfaces by inclusion maps from a fixed surface of the same genus so that the distance functions corresponding to the pullback metrics converge to a pseudo-metric and the inclusion maps converge to a Lipschitz map. We show further that the limiting pseudo-metric has fractal dimension two.

As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and genus of a surface $F\subset M$, together with bounds on the geometry of $M$, give an upper bound on the diameter of $F$.

Our proof is modelled on Gromov’s compactness theorem for $J$-holomorphic curves.

References

William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491. MR 307015, DOI 10.2307/1970868
Michael T. Anderson, Curvature estimates for minimal surfaces in $3$-manifolds, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 1, 89–105. MR 803196
Hyeong In Choi and Richard Schoen, The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature, Invent. Math. 81 (1985), no. 3, 387–394. MR 807063, DOI 10.1007/BF01388577
Tobias H. Colding and William P. Minicozzi II, Convergence of embedded minimal surfaces without area bounds in three-manifolds, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 8, 765–770 (English, with English and French summaries). MR 1659982, DOI 10.1016/S0764-4442(98)80167-5
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
Joel Hass and Peter Scott, The existence of least area surfaces in $3$-manifolds, Trans. Amer. Math. Soc. 310 (1988), no. 1, 87–114. MR 965747, DOI 10.1090/S0002-9947-1988-0965747-6
Christoph Hummel, Gromov’s compactness theorem for pseudo-holomorphic curves, Progress in Mathematics, vol. 151, Birkhäuser Verlag, Basel, 1997. MR 1451624, DOI 10.1007/978-3-0348-8952-0
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
Sumner Byron Myers, Connections between differential geometry and topology. I. Simply connected surfaces, Duke Math. J. 1 (1935), no. 3, 376–391. MR 1545884, DOI 10.1215/S0012-7094-35-00126-0
Sumner Byron Myers, Connections between differential geometry and topology II. Closed surfaces, Duke Math. J. 2 (1936), no. 1, 95–102. MR 1545908, DOI 10.1215/S0012-7094-36-00208-9
Peter Topping, Relating diameter and mean curvature for submanifolds of Euclidean space, Comment. Math. Helv. 83 (2008), no. 3, 539–546. MR 2410779, DOI 10.4171/CMH/135
B. White, Curvature estimates and compactness theorems in $3$-manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math. 88 (1987), no. 2, 243–256. MR 880951, DOI 10.1007/BF01388908
J. G. Wolfson, Gromov’s compactness of pseudo-holomorphic curves and symplectic geometry, J. Differential Geom. 28 (1988), no. 3, 383–405. MR 965221