On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. We prove that this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary, as motivated by an attempt to generalize the Riemannian Penrose inequality in dimension 8. This result is a generalization of Corvino's result about localized scalar curvature deformations; however, the existence part needs to be handled delicately since the problem is non-variational. For non-generic cases, we give a classification theorem for domains in space forms and Schwarzschild manifolds, and show the connection with positive mass theorems.

This work investigates two regularization techniques designed for identifying critical points of the Yang-Mills energy.

In the first half of the dissertation, we define a family of higher order functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for critical dimensions. Consequently, we generalize the results of the convergence of Yang-Mills flow in dimensions 2 and 3 given by Rade and the bubbling criterion in dimension 4 of Struwe in the case where the initial flow data is smooth. This encompasses the contents of the author’s paper solo paper from 2014.

In the second half of the dissertation we study an alternate type of regularization. In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify Yang-Mills connections obtained by approximations with respect to the Yang-Mills α-energy. More specifically, we show that for the SU(2) Hopf fibration over S4, for sufficiently small α values the SO(4)-invariant ADHM instanton is the unique α-critical point which has Yang-Mills α-energy lower than a specific threshold. This is an overview of the author’s solo paper from 2016.

In 1978, the physicist P.S. Jang introduced a quasilinear elliptic equation in an attempt to generalize Geroch's approach to the positive mass conjecture of general relativity. The first existence and regularity result of Jang's equation was obtained by R. Schoen and S.-T. Yau through the capillary regularization procedure and stability-based a priori estimates. Yet, the solutions produced by this procedure may blow up in some black hole regions.

Schoen--Yau showed that the graph of a blowup solution to Jang's equation is asymptotic to cylinders over apparent horizons. J. Metzger showed that such cylindrical asymptotics are exponential, and he estimated the asymptotic rate by certain spectral properties of apparent horizons, followed by Q. Han, M. Khuri, and W. Yu. Their estimates involve delicate barrier construction and require the assistance of regularized solutions. We provide a simple proof of the sharp estimates that also apply to general blowup solutions.

We prove the first analytic and geometric result of regularized solutions to Jang's equation in black hole regions by applying two natural geometric treatments: translation and dilation. First, we show that the graphs of properly translated solutions converge subsequentially to constant expansion surfaces. Second, we characterize the limits of properly rescaled solutions. Third, we investigate the structure of black hole regions that arise in the Schoen--Yau regularization procedure. Finally, we discuss a special case of low-speed blowup behavior.