Lu Wang (汪璐) - Research

Research Interests

I primarily work in differential geometry, geometric analysis and partial differential equations - more specifically, geometric flows (mean curvature flow, Ricci flow, and harmonic map heat flow) and minimal surfaces. I am also interested in other related topics, such as low-dimensional topology.

Geometric flows are the gradient flows associated to functionals on manifolds which have geometric interpretations. Not only are geometric flows of great interest in several branches of mathematics, such as differential geometry and topology, nonlinear PDEs and calculus of variations, but they also have potential applications to questions arising from other scientific fields, such as biology, computer imaging, material sciences and physics.

My recent work has focused on geometric and analytic properties of self-similar solutions (also called "solitons") to the mean curvature flow and Ricci flow, which often arise as models of singularities of the flow. I am interested in understanding the asymptotic structure as well as uniqueness of these solutions. Currently I am investigating the role of Colding-Minicozzi's notion of entropy in the study of these questions.

Journal Publications

A mountain-pass theorem for asymptotically conical self-expanders (with J. Bernstein), accepted by Peking Math. J. (2021). ArXiv.
Closed hypersurfaces of low entropy in $\mathbb{R}^4$ are isotopically trivial (with J. Bernstein), accepted by Duke Math. J. (2021). ArXiv.
A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons (with B. Kotschwar), accepted by J. Differential Geom. (2020). ArXiv.
The space of asymptotically conical self-expanders of mean curvature flow (with J. Bernstein), Math. Ann. 380 (2021), no. 1-2, 175-230. Journal; ArXiv.
Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow (with J. Bernstein), Int. Math. Res. Not. IMRN 2021, no. 12, 9016-9044. Journal; ArXiv.
Hausdorff stability of the round two-sphere under small perturbations of the entropy (with J. Bernstein), Math. Res. Lett. 25 (2018), no. 2, 347-365. Journal; ArXiv.
Topology of closed hypersurfaces of small entropy (with J. Bernstein), Geom. Topol. 22 (2018), no. 2, 1109-1141. Journal; ArXiv.
A topological property of asymptotically conical self-shrinkers of small entropy (with J. Bernstein), Duke Math. J. 166 (2017), no. 3, 403-435. Journal; ArXiv.
A sharp lower bound for the entropy of closed hypersurfaces up to dimension six (with J. Bernstein), Invent. Math. 206 (2016), no. 3, 601-627. Journal; ArXiv.
Uniqueness of self-similar shrinkers with asymptotically cylindrical ends, J. Reine Angew. Math. 715 (2016), 207-230. Journal; PDF.
A remark on a uniqueness property of high multiplicity tangent flows in dimension 3 (with J. Bernstein), Int. Math. Res. Not. IMRN 2015, no. 15, 6286-6294. Journal; ArXiv.
Rigidity of asymptotically conical shrinking gradient Ricci solitons (with B. Kotschwar), J. Differential Geom. 100 (2015), no. 1, 55-108. Journal; ArXiv.
Uniqueness of self-similar shrinkers with asymptotically conical ends, J. Amer. Math. Soc. 27 (2014), no. 3, 613-638. Journal; ArXiv.
Harmonic map heat flow with rough boundary data, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5265-5283. Journal; ArXiv.
A Bernstein type theorem for self-similar shrinkers, Geom. Dedicata 151 (2011), 297-303. Journal Link; ArXiv.
Existence of good sweepouts on closed manifolds (with L. Lin), Proc. Amer. Math. Soc. 138 (2010), 4081-4088. Journal Link; ArXiv.
A quartic system and a quintic system with fine focus of order 18 (with J. Huang, F. Wang, and J. Yang), Bull. Sci. Math. 132 (2008), no. 3, 205-217. Journal; PDF.

Preprints

Asymptotic structure of self-shrinkers, preprint (2016). ArXiv.
An integer degree for asymptotically conical self-expanders (with J. Bernstein), preprint (2018). ArXiv.
Isometries of asymptotically conical shrinking Ricci solitons (with B. Kotschwar), preprint (2019). ArXiv.
Topological uniqueness for self-expanders of small entropy (with J. Bernstein), preprint (2019). ArXiv.
Relative expander entropy in the presence of a two-sided obstacle and applications (with J. Bernstein), preprint (2019). ArXiv.

Conference Reports, Surveys, etc.

Entropy in mean curvature flow, to appear in Proceedings of the International Congress of Mathematicians 2022. PDF.
The space of asymptotically conical self-expanders of mean curvature flow (extended abstract), Oberwolfach Reports (2018). MFO; PDF.
Mean curvature flow and entropy (extended abstract), Oberwolfach Reports (2017). MFO; PDF.
Asymptotic structure of self-shrinkers of mean curvature flow (extended abstract), Oberwolfach Reports (2017). MFO; PDF.