Research

Research Interests

Broadly speaking, my research is in geometric analysis: conformal geometry, partial differential equations, the calculus of variations, and global differential geometry.

Recent Preprints and Publications:

X. Cao, M. J. Gursky, and H. Tran, "Curvature of the Second kind and a conjecture of Nishikawa", https://arxiv.org/abs/2112.01212

M. J. Gursky and S. Perez-Ayala, " Variational Properties Of The Second Eigenvalue Of The Conformal Laplacian," to appear in J. Funct. Anal., https://arxiv.org/abs/2010.13210

M. J. Gursky, S. McKeown, and A. Tyrrell, "Renormalized volume of minimally bounded regions in asymptotically hyperbolic Einstein spaces," https://arxiv.org/abs/2101.11755

M. J. Gursky and S. Zhang, "Rigidity for Bach-flat metrics on manifolds with boundary and applications", https://arxiv.org/abs/2007.04355

M. J. Gursky, Q. Han, and S. Stolz, "An invariant related to the existence of conformally compact Einstein fillings", Trans. Amer. Math. Soc. 374 (2021), no. 6, 4185–4205, https://arxiv.org/abs/1801.04474

C. R. Graham and M. J. Gursky, "Chern-Gauss-Bonnet formula for singular Yamabe metrics in dimension four," Indiana Univ. Math. J. 70 (2021), no. 3, 1131–1166, https://arxiv.org/abs/1902.01562

M. J. Gursky, C. Kelleher, and J. Streets, "Index-energy estimates for Yang-Mills connections and Einstein metrics’’, Comm. Math. Phys. 376 (2020), no. 1, 117–143, http://arxiv.org/abs/1901.04539

S.Y.A Chang, M.J Gursky, and S. Zhang," A conformally invariant gap theorem characterizing CP2 via the Ricci flow", Math. Z. 294 (2020), no. 1-2, 721–746, https://arxiv.org/abs/1809.05918

M. J. Gursky and G. Szekelyhidi, " A Local Existence Result for Poincare-Einstein metrics", Adv. Math. 361 (2020), https://arxiv.org/abs/1712.04017

M. J. Gursky and J. Streets, "A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow’’, Geometric Flows, 4 (2019), no. 1, 30-50, https://arxiv.org/abs/1507.04781

M. J. Gursky and J. Streets, "Variational structure of the $v_{\frac{n}{2}}$-Yamabe problem’’, Differential Geom. Appl. 56 (2018), 187–201. https://www.sciencedirect.com/science/article/pii/S0926224517301122

M. J. Gursky, C. Kelleher, and J. Streets, "A conformally invariant gap theorem in Yang-Mills theory’’, Comm. Math. Physics, 361 (2018), 1155-1167 https://link.springer.com/article/10.1007/s00220-017-3070-z

M. J. Gursky and J. Streets, "A formal Riemannian structure on conformal classes and uniqueness for the $\sigma_{2}$-Yamabe problem’’, Geom. & Topology. 22 (2018), 3501-3573 https://arxiv.org/abs/1603.07005

M. J. Gursky and Q. Han, "Non-existence of Poincaré-Einstein manifolds with prescribed conformal infinity’’, Geom. Funct. Anal. 27 (2017), no. 4, 863--879. https://link-springer-com.proxy.library.nd.edu/article/10.1007%2Fs00039-017-0414-y