Research

My primary field of interest is geometric analysis - with current emphasis on constant mean curvature surfaces, minimal surfaces, and harmonic maps.

Publications:

• Conservation Laws and Gluing Constructions for Constant Mean Curvature (Hyper)Surfaces, Joint with N. Kapouleas and S. Kleene, "Notices of the AMS", May 2022.

• Harmonic branched coverings and uniformization of CAT(k) spheres, Joint with C. Mese, "J. Reine Angew. Math." (Crelle's Journal), 779 (2021), 123–166.

• Complete Constant Mean Curvature Hypersurfaces in Euclidean space of dimension four or higher , Joint with N. Kapouleas, "Amer. J. Math." (2021), no. 4, 1161-1259.

• Bubble tree convergence for harmonic maps into compact locally CAT(1) spaces , Joint with S, Lakzian, "Calc. Var. and PDE", 59 (2020), no. 5, 23 pp.

• Existence of harmonic maps into CAT(1) spaces , Joint with A. Fraser, L.-H. Huang, C. Mese, P. Sargent, Y. Zhang, "Comm. Anal. Geom." 28 (2020), no. 4, 781-835.

• Logarithmically spiraling helicoids, Joint with S. Kleene, "Comm. Anal. Geom." 26 (2018), no. 3, 461-484.

• Regularity of harmonic maps from polyhedra to CAT(1) spaces , Joint with A. Fraser, L.-H. Huang, C. Mese, P. Sargent, Y. Zhang, "Calc. Var. and PDE" 57 (2018) no. 1.

• Quantitative stratification and higher regularity for biharmonic maps, Joint with T. Lamm, "Manuscripta Math." 148 (2015) 379-398.

• Compactness results for sequences of approximate biharmonic maps, Joint with T. Lamm, ''Pacific J. Math." 276 (2015) no. 1, 59-92.

• A minimal lamination of the interior of a positive cone with quadratic curvature blowup, Joint with S. Kleene, "J. Geom. Anal." 25 (2015) 1409-1420

• Embedded Constant Mean Curvature Surfaces in Euclidean Three Space, Joint with N. Kapouleas, "Math. Ann." 360 (2014) 1041-1108.

• A Variational Characterization of the Catenoid, Joint with J. Bernstein, "Calc. Var. and PDE" 49 (2014) 215-232.

• Symmetry of Embedded Genus-One Helicoids, Joint with Jacob Bernstein, "Duke Math. J." 159 (2011) 83-97.

• Conformal Structure of Minimal Surfaces with Finite Topology, Joint with Jacob Bernstein, "Comm. Math. Helv." 86 (2011) 353-381.

• Helicoid-like Minimal Disks and Uniqueness, Joint with Jacob Bernstein, "J. Reine Angew. Math." (Crelle's Journal) 655 (2011) 129-146.

• Distortions of the Helicoid, Joint with Jacob Bernstein, "Geometriae Dedicata." 137 (1), (2008) 143-147.