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

• Ancient pancake solutions to fully nonlinear curvature flows. With Sathya Rengaswami.
• A distance comparison principle for curve shortening flow with free boundary. With Jonathan Zhu.
• Ancient Ricci flows of bounded girth. With Theodora Bourni, Tim Buttsworth and Ramiro Lafuente.
• Classification of convex ancient free boundary mean curvature flows in the ball. With Theodora Bourni.
• Collapsing and noncollapsing in convex ancient mean curvature flow. With Theodora Bourni and Stephen Lynch.
• Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery. With Stephen Lynch and Huy The Nguyen.

Survey articles

• Translating solutions to mean curvature flow. In Minimal Surfaces: Integrable Systems and Visualisation, 1–12, Springer Proc. Math. Stat.. Vol. 349 (2021). With Theodora Bourni and Giuseppe Tinaglia. [Preprint version]
• Convex ancient solutions to mean curvature flow. In Differential Geometry in the Large, London Mathematical Society Lecture Note Series. Cambridge University Press. Vol. 463 (2020) With Theodora Bourni and Giuseppe Tinaglia. [Preprint version]
• Ancient solutions to mean curvature flow. With Theodora Bourni and Giuseppe Tinaglia.

Thesis

• Motion of hypersurfaces by curvature. Australian National University, Canberra (2015) [ANU open access theses]