Research Interests

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

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Published Articles

• Classification of convex ancient free boundary curve shortening flows in the disc. Classification of convex ancient free boundary curve shortening flows in the disc, Analysis and PDE (to appear). With Mat Langford.
• CMC hypersurfaces with bounded Morse index. 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 $\mathbb R^2$. 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 and Topology (to appear). With Mat Langford and Giuseppe Tinaglia.
• Differential Harnack inequalities via concavity of the arrival time. Communications in Analysis and Geometry (to appear). 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 f\"ur 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

• Nonplanar ancient curve shortening flows in $\mathbb R^3$ from grim reapers. With Alex Mramor.
• Collapsing and noncollapsing in convex ancient mean curvature flow. With Mat Langford and Stephen Lynch.

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.