Research interests
Ricci flow and flows of hypersurfaces by (nonlinear) functions of their curvature, related nonlinear elliptic and parabolic PDE, and applications to topology, geometry and general relativity.
Publications
Books
- (Author, with Ben Andrews, Bennett Chow and Christine Guenther.) Extrinsic Geometric Flows. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI. Vol. 206 (2020), 790pp. [Preview material]
- (Editor, with Theodora Bourni.) Mean Curvature Flow: Proceedings of the John H. Barrett Memorial Lectures Held at the University of Tennessee, Knoxville, May 29 — June 1, 2018. De Gruyter Proceedings in Mathematics, De Gruyter, Berlin (2020), 232pp.
Refereed articles
- Sharp pinching estimates for mean curvature flow in the sphere. To appear in Commun. Anal. Geom. With Huy The Nguyen. [Prepeint version]
- Classification of convex ancient free boundary curve shortening flows in the disc. To appear in Anal. PDE. With Theodora Bourni. [Preprint version]
- Differential Harnack inequalities via concavity of the arrival time. To appear in Commun. Anal. Geom. With Theodora Bourni. [Preprint version]
- Local convexity estimates for mean curvature flow. J. Reine Angew. Math. (ahead of print) [Preprint version]
- Pinched hypersurfaces are compact. Adv. Nonlinear Stud. 23 (2023) no. 1, ans–2022–0046. With Theodora Bourni and Stephen Lynch. [Preprint version]
- Ancient mean curvature flows out of polytopes. Geom. Topol. 26, no. 4 (2022), 1849–1905. With Theodora Bourni and Giuseppe Tinaglia. [Preprint version (note change of title)]
- Collapsing ancient solutions of mean curvature flow. J. Differential Geom. 119, no. 2 (2021), pp. 187–219. With Theodora Bourni and Giuseppe Tinaglia. [Preprint version] (Click here for a brief description of the 'ancient pancake' solution, including some motion pictures.)
- Quadratically pinched hypersurfaces of the sphere via mean curvature flow with surgery. Calc. Var. Partial Differential Equations. 60, 216 (2021) With Huy The Nguyen. [Preprint version]
- Concavity of solutions to degenerate elliptic equations on the sphere. Comm. Partial Differential Equations. 46 (2021), no. 6, 1005–1016. With Julian Scheuer. [Preprint version]
- On the construction of closed nonconvex nonsoliton ancient mean curvature flows. Int. Math. Res. Not. (2021), no. 1, 757–768. With Theodora Bourni and Alex Mramor. [Preprint version]
- Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions. J. Reine Angew. Math. 765 (2020), 1–33 With Stephen Lynch. [Preprint version]
- Convex ancient solutions to curve shortening flow. Calc. Var. Partial Differential Equations 59 (2020), no. 4, 133. With Theodora Bourni and Giuseppe Tinaglia. [Preprint version]
- On the existence of convex translators in slab regions. Anal. PDE 13 (2020), no. 4, 1051–1072. With Theodora Bourni and Giuseppe Tinaglia. [Preprint version]
- A general pinching principle for mean curvature flow and applications. Calc. Var. Partial Differential Equations 56 (2017), no. 4, paper no. 107, 31 pp. [Preprint version]
- Type-II singularities of two-convex immersed mean curvature flow. Geom. Flows 2 (2016), no. 1, 1–17. With Theodora Bourni. [Open access]
- Two-sided non-collapsing curvature flows. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 15 (2016), 543–560. With Ben Andrews. [Preprint version]
- The optimal interior ball estimate for a k-convex mean curvature flow. Proc. Am. Math. Soc. 143 (2015), no. 12, 5395–5398. [Preprint version]
- Convexity estimates for surfaces moving by curvature functions. J. Differential Geom. 99 (2015), no. 1, 47–75. With Ben Andrews and James McCoy. [Preprint version]
- Cylindrical estimates for hypersurfaces moving by convex curvature functions. Anal. PDE 7 (2014), no. 5, 1091–1107. With Ben Andrews. [Preprint version]
- Convexity estimates for hypersurfaces moving by convex curvature functions. Anal. PDE 7 (2014), no. 2, 407–433. With Ben Andrews and James McCoy. [Preprint version]
- Non-collapsing in fully nonlinear curvature flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 1, 23–32. With Ben Andrews and James McCoy. [Preprint version]
Preprints
Survey articles
Thesis
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