Abstract
We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disc. The surfaces are constructed by singular perturbation methods and have three boundary components. They are the free boundary analogue of the Costa–Hoffman–Meeks surfaces and the surfaces constructed by Kapouleas by desingularizing coaxial catenoids and planes. It is plausible that the minimal surfaces we constructed here are the same as the ones obtained recently by Ketover by using the min-max method.
Funding statement: Martin Li would like to thank the Croucher Foundation for the financial support and the Department of Mathematics at Massachusetts Institute of Technology, where part of the work in this paper was done. He was partially supported by CUHK Direct Grant for Research C001-4053118 and a grant from the Research Grants Council of the Hong Kong SAR, China (Project No. CUHK 24305115). Nikolaos Kapouleas was partially supported by NSF grants DMS-1105371 and DMS-1405537.
A Local exponential map estimates
Proposition A.1.
Let g be a Riemannian metric on
for some constant
and that the exponential map
where
Proof.
The inverse of a matrix A is given by
where
By standard ODE theory and the estimates in (A.3), the exponential map
It remains to prove (A.4). The smoothness of the exponential map is a direct consequence of the smooth dependence on initial conditions
Let
Using (A.2) and
By Taylor’s theorem,
For estimates on the derivatives, we differentiate the system (A.5). For example, differentiating with respect to some
Recall Kato’s inequality that
If we define the function
then G is a nonnegative monotone increasing function hence differentiable a.e. and from (A.6), we have the differential inequality
with
provided
The estimate on the derivatives with respect to v can be obtained similarly. We differentiate (A.5) with respect to some
Define
Higher-order derivative estimates can be obtained in a similar manner. ∎
As a corollary of the above exponential map estimates, one can prove the following lower bound on the injectivity radius.
Corollary A.8.
Under the same assumption as in Proposition A.1, we have
for all
Proof.
Since
Acknowledgements
The authors would like to thank Richard Schoen for his continuous support and interest in the results of this article. They would like to thank also the referee for carefully reading the manuscript and making many helpful suggestions.
References
[1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. 10.1002/cpa.3160120405Search in Google Scholar
[2] E. G. Björling, In integrationem aequationis derivatarum partialum superfici, cujus in puncto unoquoque principales ambo radii curvedinis aequales sunt signoque contrario, Arch. Math. Phys. 4 (1844), 290–315. Search in Google Scholar
[3] J. Chen, A. Fraser and C. Pang, Minimal immersions of compact bordered Riemann surfaces with free boundary, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2487–2507. 10.1090/S0002-9947-2014-05990-4Search in Google Scholar
[4]
J. Choe and M. Soret,
New minimal surfaces in
[5]
C. J. Costa,
Example of a complete minimal immersion in
[6] R. Courant, The existence of minimal surfaces of given topological structure under prescribed boundary conditions, Acta Math. 72 (1940), 51–98. 10.1007/BF02546328Search in Google Scholar
[7] S. K. Donaldson, Connections, cohomology and the intersection forms of 4-manifolds, J. Differential Geom. 24 (1986), no. 3, 275–341. 10.4310/jdg/1214440551Search in Google Scholar
[8] A. Folha, F. Pacard and T. Zolotareva, Free boundary minimal surfaces in the unit 3-ball, Manuscripta Math. 154 (2017), no. 3–4, 359–409. 10.1007/s00229-017-0924-9Search in Google Scholar
[9] R. L. Foote, Regularity of the distance function, Proc. Amer. Math. Soc. 92 (1984), no. 1, 153–155. 10.1090/S0002-9939-1984-0749908-9Search in Google Scholar
[10] A. Fraser, On the free boundary variational problem for minimal disks, Comm. Pure Appl. Math. 53 (2000), no. 8, 931–971. 10.1002/1097-0312(200008)53:8<931::AID-CPA1>3.0.CO;2-9Search in Google Scholar
[11] A. Fraser, Minimal disks and two-convex hypersurfaces, Amer. J. Math. 124 (2002), no. 3, 483–493. 10.1353/ajm.2002.0013Search in Google Scholar
[12] A. Fraser, Index estimates for minimal surfaces and k-convexity, Proc. Amer. Math. Soc. 135 (2007), no. 11, 3733–3744. 10.1090/S0002-9939-07-08894-6Search in Google Scholar
[13] A. Fraser and M. M.-C. Li, Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary, J. Differential Geom. 96 (2014), no. 2, 183–200. 10.4310/jdg/1393424916Search in Google Scholar
[14] A. Fraser and R. Schoen, The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math. 226 (2011), no. 5, 4011–4030. 10.1016/j.aim.2010.11.007Search in Google Scholar
[15] A. Fraser and R. Schoen, Uniqueness theorems for free boundary minimal disks in space forms, Int. Math. Res. Not. IMRN 2015 (2015), no. 17, 8268–8274. 10.1093/imrn/rnu192Search in Google Scholar
[16] A. Fraser and R. Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. Math. 203 (2016), no. 3, 823–890. 10.1007/s00222-015-0604-xSearch in Google Scholar
[17] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Class. Math., Springer, Berlin 2001. 10.1007/978-3-642-61798-0Search in Google Scholar
[18] M. Grüter and J. Jost, On embedded minimal disks in convex bodies, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 5, 345–390. 10.1016/s0294-1449(16)30378-xSearch in Google Scholar
[19] S. Hildebrandt and J. C. C. Nitsche, Minimal surfaces with free boundaries, Acta Math. 143 (1979), no. 3–4, 251–272. 10.1007/BF02392096Search in Google Scholar
[20]
D. Hoffman and W. H. Meeks, III,
A complete embedded minimal surface in
[21] D. Hoffman and W. H. Meeks, III, A variational approach to the existence of complete embedded minimal surfaces, Duke Math. J. 57 (1988), no. 3, 877–893. 10.1215/S0012-7094-88-05739-0Search in Google Scholar
[22] D. Hoffman and W. H. Meeks, III, Embedded minimal surfaces of finite topology, Ann. of Math. (2) 131 (1990), no. 1, 1–34. 10.2307/1971506Search in Google Scholar
[23] D. Hoffman and W. H. Meeks, III, Limits of minimal surfaces and Scherk’s fifth surface, Arch. Ration. Mech. Anal. 111 (1990), no. 2, 181–195. 10.1007/BF00375407Search in Google Scholar
[24] D. Hoffman and W. H. Meeks, III, Minimal surfaces based on the catenoid, Amer. Math. Monthly 97 (1990), no. 8, 702–730. 10.1080/00029890.1990.11995658Search in Google Scholar
[25] J. Jost, Existence results for embedded minimal surfaces of controlled topological type. I, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 13 (1986), no. 1, 15–50. Search in Google Scholar
[26] N. Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131 (1990), no. 2, 239–330. 10.2307/1971494Search in Google Scholar
[27] N. Kapouleas, Compact constant mean curvature surfaces in Euclidean three-space, J. Differential Geom. 33 (1991), no. 3, 683–715. 10.4310/jdg/1214446560Search in Google Scholar
[28] N. Kapouleas, Constant mean curvature surfaces constructed by fusing Wente tori, Proc. Natl. Acad. Sci. USA 89 (1992), no. 12, 5695–5698. 10.1073/pnas.89.12.5695Search in Google Scholar PubMed PubMed Central
[29] N. Kapouleas, Constant mean curvature surfaces constructed by fusing Wente tori, Invent. Math. 119 (1995), no. 3, 443–518. 10.1007/BF01245190Search in Google Scholar
[30] N. Kapouleas, Complete embedded minimal surfaces of finite total curvature, J. Differential Geom. 47 (1997), no. 1, 95–169. 10.4310/jdg/1214460038Search in Google Scholar
[31] N. Kapouleas, Constructions of minimal surfaces by gluing minimal immersions, Global theory of minimal surfaces, Clay Math. Proc. 2, American Mathematical Society, Providence (2005), 489–524. Search in Google Scholar
[32] N. Kapouleas, Doubling and desingularization constructions for minimal surfaces, Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM) 20, International Press, Somerville (2011), 281–325. Search in Google Scholar
[33] N. Kapouleas, Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere. I, J. Differential Geom. 106 (2017), no. 3, 393–449. 10.4310/jdg/1500084022Search in Google Scholar
[34] N. Kapouleas, A general desingularization theorem for minimal surfaces in the compact case, in preparation. Search in Google Scholar
[35] N. Kapouleas, S. J. Kleene and N. M. Mø ller, Mean curvature self-shrinkers of high genus: Non-compact examples, J. reine angew. Math. 739 (2018), 1–39. 10.1515/crelle-2015-0050Search in Google Scholar
[36] N. Kapouleas and P. McGrath, Minimal surfaces in the round three-sphere by doubling the equatorial two-sphere. II, Comm. Pure Appl. Math. 72 (2019), no. 10, 2121–2195. 10.1002/cpa.21854Search in Google Scholar
[37] N. Kapouleas and D. Wiygul, Minimal surfaces in the 3-sphere by desingulariziang intersecting Clifford tori, preprint (2017), https://arxiv.org/abs/1701.05658. Search in Google Scholar
[38] N. Kapouleas and S.-D. Yang, Minimal surfaces in the three-sphere by doubling the Clifford torus, Amer. J. Math. 132 (2010), no. 2, 257–295. 10.1353/ajm.0.0104Search in Google Scholar
[39]
H. Karcher, U. Pinkall and I. Sterling,
New minimal surfaces in
[40] D. Ketover, Free boundary minimal surfaces of unbounded genus, preprint (2016), https://arxiv.org/abs/1612.08691. Search in Google Scholar
[41]
H. B. Lawson, Jr.,
Complete minimal surfaces in
[42] M. M.-C. Li, A general existence theorem for embedded minimal surfaces with free boundary, Comm. Pure Appl. Math. 68 (2015), no. 2, 286–331. 10.1002/cpa.21513Search in Google Scholar
[43] M. M.-C. Li and X. Zhou, Min-max theory for free boundary minimal hypersurfaces I: Regularity theory, preprint (2016), https://arxiv.org/abs/1611.02612. 10.4310/jdg/1625860624Search in Google Scholar
[44] D. Maximo, I. Nunes and G. Smith, Free boundary minimal annuli in convex three-manifolds, J. Differential Geom. 106 (2017), no. 1, 139–186. 10.4310/jdg/1493172096Search in Google Scholar
[45] W. Meeks, III, L. Simon and S. T. Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. 10.2307/2007026Search in Google Scholar
[46] S. Montiel and A. Ros, Schrödinger operators associated to a holomorphic map, Global differential geometry and global analysis (Berlin 1990), Lecture Notes in Math. 1481, Springer, Berlin (1991), 147–174. 10.1007/BFb0083639Search in Google Scholar
[47] X. H. Nguyen, Construction of complete embedded self-similar surfaces under mean curvature flow. Part III, Duke Math. J. 163 (2014), no. 11, 2023–2056. 10.1215/00127094-2795108Search in Google Scholar
[48] J. C. C. Nitsche, The regularity of the trace for minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 3 (1976), no. 1, 139–155. Search in Google Scholar
[49] J. C. C. Nitsche, Stationary partitioning of convex bodies, Arch. Ration. Mech. Anal. 89 (1985), no. 1, 1–19. 10.1007/BF00281743Search in Google Scholar
[50] S. Pigola and G. Veronelli, The smooth Riemannian extension problem, preprint (2016), https://arxiv.org/abs/1606.08320. 10.2422/2036-2145.201802_013Search in Google Scholar
[51] H. F. Scherk, Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen, J. reine angew. Math. 13 (1835), 185–208. 10.1515/crll.1835.13.185Search in Google Scholar
[52] R. Schoen and S. T. Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. 10.2307/1971247Search in Google Scholar
[53] R. Schoen and S. T. Yau, Complete three-dimensional manifolds with positive Ricci curvature and scalar curvature, Seminar on differential geometry, Ann. of Math. Stud. 102, Princeton University, Princeton (1982), 209–228. 10.1515/9781400881918-013Search in Google Scholar
[54] R. M. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure Appl. Math. 41 (1988), no. 3, 317–392. 10.1002/cpa.3160410305Search in Google Scholar
[55] C. H. Taubes, Self-dual Yang–Mills connections on non-self-dual 4-manifolds, J. Differential Geometry 17 (1982), no. 1, 139–170. 10.4310/jdg/1214436701Search in Google Scholar
[56] C. H. Taubes, Path-connected Yang–Mills moduli spaces, J. Differential Geom. 19 (1984), no. 2, 337–392. 10.4310/jdg/1214438683Search in Google Scholar
[57] C. H. Taubes, A framework for Morse theory for the Yang–Mills functional, Invent. Math. 94 (1988), no. 2, 327–402. 10.1007/BF01394329Search in Google Scholar
[58] J. E. Taylor, Boundary regularity for solutions to various capillarity and free boundary problems, Comm. Partial Differential Equations 2 (1977), no. 4, 323–357. 10.1080/03605307708820033Search in Google Scholar
[59] M. Traizet, Construction de surfaces minimales en recollant des surfaces de Scherk, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 5, 1385–1442. 10.5802/aif.1554Search in Google Scholar
[60] H. C. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), no. 1, 193–243. 10.2140/pjm.1986.121.193Search in Google Scholar
[61] D. Wiygul, Minimal surfaces in the 3-sphere by stacking Clifford tori, J. Differential Geom. 114 (2020), no. 3, 467–549. 10.4310/jdg/1583377214Search in Google Scholar
[62] M. Wohlgemuth, Higher genus minimal surfaces by growing handles out of a catenoid, Manuscripta Math. 70 (1991), no. 4, 397–428. 10.1007/BF02568387Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
