Research Interests

Geometric analysis, differential geometry, partial differential equations, geometric flows, and geometric measure theory.

Published Articles

• Convex Ancient Solutions to Anisotropic Curve Shortening Flow. Revista Matematica Iberoamericana, 40 (2024), no. 6, pp. 2311--2324. With Benjamin Richards.
• Compact curve shortening flow solutions out of non compact curve. International Mathematics Research Notices, Volume 2024, Issue 1, January 2024, Pages 804--816. With Martin Reiris.
• Nonplanar ancient curve shortening flows in $\mathbb R^3$ from grim reapers. Mathematische Zeitschrift, to appear. With Alex Mramor.
• Collapsing and noncollapsing in convex ancient mean curvature flow. Journal fur Reine und Angewandte Mathematik, no. 801, 2023, pp. 273--305. With Mat Langford and Stephen Lynch.
• Classification of convex ancient free boundary mean curvature flows in the ball. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, to appear. With Mat Langford.
• Pinched hypersurfaces are compact. Advanced Nonlinear Studies, vol. 23, no. 1, 2023, pp. 20220046. With Mat Langford and Stephen Lynch.
• Classification of convex ancient free boundary curve shortening flows in the disc. Analysis and PDE, 16-9 (2023), 2225-2240. With Mat Langford.
• CMC hypersurfaces with bounded Morse index. Journal fur die Reine und Angewandte Mathematik, 786 (2022), 175–203. With Ben Sharp and Giuseppe Tinaglia.
• Ancient solutions for flow by powers of the curvature in $\mathbb R^2$. Calculus of Variations and Partial Differential Equations, 61 (2022), no. 2, Paper No. 44. With Julie Clutterbuck, Xuan Hien Nguyen, Alina Stancu, Guofang Wei and Valentina-Mira Wheeler.
• The vanishing of the fundamental gap of convex domains in $\mathbb H^n$. Annales Henri Poincare, 23, 595--614 (2022). With Julie Clutterbuck, Xuan Hien Nguyen, Alina Stancu, Guofang Wei and Valentina-Mira Wheeler.
• Ancient solutions to mean curvature flow out of polygons. Geometry & Topology 26(4), 1849-1905 (2022). With Mat Langford and Giuseppe Tinaglia.
• Differential Harnack inequalities via concavity of the arrival time. Communications in Analysis and Geometry, Vol. 31, No. 3 (2023), pp. 547-561. With Mat Langford.
• Collapsing ancient solutions of mean curvature flow. Journal of Differential Geometry, 119, No. 2 (2021), pp. 187–219. With Mat Langford and Giuseppe Tinaglia.
• Explicit fundamental gap estimates for some convex domains in $\mathbb H^2$’’. Mathematical Research Letters, Volume 28, Number 5, 1319--1336, 2021 . With Julie Clutterbuck, Xuan Hien Nguyen, Alina Stancu, Guofang Wei and Valentina-Mira Wheeler.
• On the construction of closed nonconvex nonsoliton ancient mean curvature flows. International Mathematical Research Notices, Volume 2021, Issue 1, January 2021, Pages 754--765. With Mat Langford and Alexander Mramor.
• Convex ancient solutions to curve shortening flow. Calculus of Variations and Partial Differential Equations, no. 4, Paper No. 133, 15pp, 2020. With Mat Langford and Giuseppe Tinaglia
• Convex ancient solutions to mean curvature flow, Differential Geometry in the Large (London Mathematical Society Lecture Note Series). Cambridge University Press, Vol 463, 2020. With Mat Langford and Giuseppe Tinaglia.
• On the existence of translating solutions of mean curvature flow in slab regions. Analysis and PDE, 13-4, 1051--1072, 2020. With Mat Langford and Giuseppe Tinaglia.
• Null mean curvature flow and outermost MOTS. Journal of Differential Geometry, 111(2), 191--239, 2019. With Kristen Moore.
• Type II singularities of two convex immersed mean curvature flow. Geometric Flows, 2(1), 1--17, 2016. With Mat Langford.
• An Allard type regularity theorem for varifolds with a H\"older condition on the first variation. Calculus of Variations and Partial Differential Equations, 55(3), Art. 46, 23pp, 2016. With Alexander Volkmann.
• Allard-type boundary regularity for $C^{1,\alpha}$ boundaries. Advances in Calculus of Variations, 9(2), 143---161, 2016.
• Area minimizing surfaces in mean convex 3-manifolds. Journal fur die Reine und Angewandte Mathematik, 704, 135--167, 2015. With Baris Coskunuzer.
• $C^{1,\alpha}$-regularity for surfaces with $H$ in $L^p$. Annals of Global Analysis and Geometry, 46(2), 159--186, 2014. With Giuseppe Tinaglia.
• Density estimates for compact surfaces with total boundary curvature less than $4\pi$. Communications in Partial Differential Equations, 37(10), 1870--1886, 2012. With Giuseppe Tinaglia.
• Curvature estimates for surfaces with bounded mean curvature. Transactions of the American Mathematical Society, 364(11), 5813--5828, 2012. With Giuseppe Tinaglia.
• $C^{1,\alpha}$ theory for the prescribed mean curvature equation with Dirichlet data. Journal of Geometric Analysis, 21(4), 982--1035, 2011.

Preprints

• On the Relation between Graph Ricci Curvature and Community Structure. With Vasileios Maroulas and Sathyanarayanan Rengaswami.
• Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains. With Nathan Burns and Spencer Catron.
• Ancient Ricci pancakes of bounded girth. With Timothy Buttsworth, Ramiro Lafuente and Mat Langford.

Books

• Mean Curvature Flow: Proceedings of the John H. Barrett Memorial Lectures held at the University of Tennessee, Knoxville, May 29--June 1, 2018, editor with Mat Langford, De Gruyter, Berlin (2020), 232pp.
• Minimal Surfaces and the Bernstein Theorem. Book chapter co-authored with Giuseppe Tinaglia. Geometry in Advanced Pure Mathematics, 167--191, LTCC Adv. Math. Ser., 4, World Sci. Publ., Hackensack, NJ, 2017.