Mean curvature flow of asymptotically conical Lagrangian submanifolds

Su, Wei-Bo

doi:10.1090/tran/7946

Mean curvature flow of asymptotically conical Lagrangian submanifolds
HTML articles powered by AMS MathViewer

by Wei-Bo Su PDF

Trans. Amer. Math. Soc. 373 (2020), 1211-1242 Request permission

Abstract:

In this paper, we study Lagrangian mean curvature flow (LMCF) of asymptotically conical (AC) Lagrangian submanifolds asymptotic to a union of special Lagrangian cones. Since these submanifolds are non-compact, we establish a short-time existence theorem for AC LMCF first. Then we focus on the equivariant, almost-calibrated case and prove long-time existence and convergence results. In particular, under certain smallness assumptions on the initial data, we show that the equivariant, almost-calibrated AC LMCF converges to a Lagrangian catenoid or an Anciaux’s expander.

References

Henri Anciaux, Construction of Lagrangian self-similar solutions to the mean curvature flow in $\Bbb C^n$, Geom. Dedicata 120 (2006), 37–48. MR 2252892, DOI 10.1007/s10711-006-9082-z
Tom Begley and Kim Moore, On short time existence of Lagrangian mean curvature flow, Math. Ann. 367 (2017), no. 3-4, 1473–1515. MR 3623231, DOI 10.1007/s00208-016-1420-3
Tapio Behrndt, Generalized Lagrangian mean curvature flow in almost Calabi-Yau manifolds, Ph.D. thesis, 2011.
Tapio Behrndt, Generalized Lagrangian mean curvature flow in Kähler manifolds that are almost Einstein, Complex and differential geometry, Springer Proc. Math., vol. 8, Springer, Heidelberg, 2011, pp. 65–79. MR 2964468, DOI 10.1007/978-3-642-20300-8_{3}
Tapio Behrndt, On the Cauchy problem for the heat equation on Riemannian manifolds with conical singularities, Q. J. Math. 64 (2013), no. 4, 981–1007. MR 3151600, DOI 10.1093/qmath/has016
Alice Chaljub-Simon and Yvonne Choquet-Bruhat, Problèmes elliptiques du second ordre sur une variété euclidienne à l’infini, Ann. Fac. Sci. Toulouse Math. (5) 1 (1979), no. 1, 9–25 (French, with English summary). MR 533596, DOI 10.5802/afst.527
Bang-yen Chen, Geometry of submanifolds and its applications, Science University of Tokyo, Tokyo, 1981. MR 627323
Bing-Long Chen and Le Yin, Uniqueness and pseudolocality theorems of the mean curvature flow, Comm. Anal. Geom. 15 (2007), no. 3, 435–490. MR 2379801, DOI 10.4310/CAG.2007.v15.n3.a1
Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI; Science Press Beijing, New York, 2006. MR 2274812, DOI 10.1090/gsm/077
Anda Degeratu and Rafe Mazzeo, Fredholm theory for elliptic operators on quasi-asymptotically conical spaces, Proc. Lond. Math. Soc. (3) 116 (2018), no. 5, 1112–1160. MR 3805053, DOI 10.1112/plms.12105
Klaus Ecker and Gerhard Huisken, Mean curvature evolution of entire graphs, Ann. of Math. (2) 130 (1989), no. 3, 453–471. MR 1025164, DOI 10.2307/1971452
Klaus Ecker and Gerhard Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), no. 3, 547–569. MR 1117150, DOI 10.1007/BF01232278
Alexander Grigor’yan and Laurent Saloff-Coste, Stability results for Harnack inequalities, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 3, 825–890 (English, with English and French summaries). MR 2149405, DOI 10.5802/aif.2116
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Reese Harvey and H. Blaine Lawson Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157. MR 666108, DOI 10.1007/BF02392726
Yohsuke Imagi, Dominic Joyce, and Joana Oliveira dos Santos, Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in $\Bbb {C}^m$, Duke Math. J. 165 (2016), no. 5, 847–933. MR 3482334, DOI 10.1215/00127094-3167275
Dominic Joyce, Special Lagrangian submanifolds with isolated conical singularities. I. Regularity, Ann. Global Anal. Geom. 25 (2004), no. 3, 201–251. MR 2053761, DOI 10.1023/B:AGAG.0000023229.72953.57
Dominic Joyce, Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow, EMS Surv. Math. Sci. 2 (2015), no. 1, 1–62. MR 3354954, DOI 10.4171/EMSS/8
Dominic Joyce, Yng-Ing Lee, and Mao-Pei Tsui, Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differential Geom. 84 (2010), no. 1, 127–161. MR 2629511
Toru Kajigaya and Keita Kunikawa, Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow, J. Geom. Phys. 128 (2018), 140–168. MR 3786192, DOI 10.1016/j.geomphys.2018.02.011
Keita Kunikawa, Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature, Geom. Dedicata 201 (2019), 369–377. MR 3978549, DOI 10.1007/s10711-018-0397-3
Gary Lawlor, The angle criterion, Invent. Math. 95 (1989), no. 2, 437–446. MR 974911, DOI 10.1007/BF01393905
Haozhao Li, Convergence of Lagrangian mean curvature flow in Kähler-Einstein manifolds, Math. Z. 271 (2012), no. 1-2, 313–342. MR 2917146, DOI 10.1007/s00209-011-0865-z
Peter Li, Geometric analysis, Cambridge Studies in Advanced Mathematics, vol. 134, Cambridge University Press, Cambridge, 2012. MR 2962229, DOI 10.1017/CBO9781139105798
Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
Robert B. Lockhart and Robert C. McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 3, 409–447. MR 837256
Jason D. Lotay and Felix Schulze, Consequences of strong stability of minimal submanifolds, International Mathematics Research Notices (2018), rny095, available at /oup/backfile/content_public/journal/imrn/pap/10.1093_imrn_rny095/2/rny095.pdf.
Peng Lu, Jie Qing, and Yu Zheng, A note on conformal Ricci flow, Pacific J. Math. 268 (2014), no. 2, 413–434. MR 3227441, DOI 10.2140/pjm.2014.268.413
Stephen Marshall, Deformations of special Lagrangian submanifolds, Ph.D. thesis, 2002.
André Neves, Singularities of Lagrangian mean curvature flow: zero-Maslov class case, Invent. Math. 168 (2007), no. 3, 449–484. MR 2299559, DOI 10.1007/s00222-007-0036-3
André Neves, Finite time singularities for Lagrangian mean curvature flow, Ann. of Math. (2) 177 (2013), no. 3, 1029–1076. MR 3034293, DOI 10.4007/annals.2013.177.3.5
Yong-Geun Oh, Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds, Invent. Math. 101 (1990), no. 2, 501–519. MR 1062973, DOI 10.1007/BF01231513
Todd A. Oliynyk and Eric Woolgar, Rotationally symmetric Ricci flow on asymptotically flat manifolds, Comm. Anal. Geom. 15 (2007), no. 3, 535–568. MR 2379804, DOI 10.4310/CAG.2007.v15.n3.a4
Andreas Savas-Halilaj and Knut Smoczyk, Lagrangian mean curvature flow of Whitney spheres, Geom. Topol. 23 (2019), no. 2, 1057–1084. MR 3939057, DOI 10.2140/gt.2019.23.1057
K. Smoczyk, A canonical way to deform a Lagrangian submanifold, eprint, arxiv:dg-ga/9605005, 1996-05.
Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), no. 1-2, 243–259. MR 1429831, DOI 10.1016/0550-3213(96)00434-8
W.-B. Su, $f$-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials, arXiv e-prints (2019-01), available at 1901.00259.
R. P. Thomas and S.-T. Yau, Special Lagrangians, stable bundles and mean curvature flow, Comm. Anal. Geom. 10 (2002), no. 5, 1075–1113. MR 1957663, DOI 10.4310/CAG.2002.v10.n5.a8
C.-J. Tsai and M.-T. Wang, A strong stability condition on minimal submanifolds and its implications, arXiv e-prints (2017-10), available at 1710.00433.
Chung-Jun Tsai and Mu-Tao Wang, Mean curvature flows in manifolds of special holonomy, J. Differential Geom. 108 (2018), no. 3, 531–569. MR 3770850, DOI 10.4310/jdg/1519959625
Craig van Coevering, Examples of asymptotically conical Ricci-flat Kähler manifolds, Math. Z. 267 (2011), no. 1-2, 465–496. MR 2772262, DOI 10.1007/s00209-009-0631-7
C. Viana, A note on the evolution of the Whitney sphere along mean curvature flow, arXiv e-prints (2018-02), available at 1802.02108.