## Mean curvature flow of asymptotically conical Lagrangian submanifolds

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- 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.

## Additional Information

**Wei-Bo Su**- Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan
- Email: d03221004@ntu.edu.tw
- Received by editor(s): March 5, 2019
- Received by editor(s) in revised form: June 30, 2019
- Published electronically: October 28, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 1211-1242 - MSC (2010): Primary 53C44; Secondary 52C38
- DOI: https://doi.org/10.1090/tran/7946
- MathSciNet review: 4068262