Liam Mazurowski

Scalar Curvature Lower Bounds and Convergence in Measure

Liam Mazurowski, Xuan Yao

Assume that smooth metrics $g_{k}$ converge in measure to a smooth metric $g$ . Gromov asked the following question: if all the metrics $g_{k}$ have non-negative scalar curvature does i…

Rigidity in the Positive Mass Theorem with $C^{0}$ Decay

Liam Mazurowski, Xuan Yao

Let $g$ be a smooth metric on $R^{3}$ with non-negative scalar curvature. We show that if $g$ satisfies $∣ g (x) - g_{euc} (x) ∣ = O (∣ x ∣^{- 1 - τ})$ for some…

A Positive Mass Theorem for Continuous Metrics

Liam Mazurowski, Xuan Yao

Let $g$ be a continuous metric on $R^{3}$ which is asymptotically flat in the sense that $∣ g_{ij} (x) - δ_{ij} ∣ = O (∣ x ∣^{- τ})$ for some $τ > \frac{1}{2}$ . Fur…

Scalar curvature under weak limits of manifolds

Liam Mazurowski, Xuan Yao

We show that scalar curvature lower bounds are preserved under certain weak convergence of smooth three manifolds to a smooth limit. More precisely, suppose that $M_{k}$ and $M$ are…

Quantification of $C^{0}$ Convergence in Dimension Three

Liam Mazurowski, Xuan Yao

We address Gromov's Quantification of $C^{0}$ Convergence Conjecture in dimension three. Let $B$ be the unit ball in $R^{3}$ . Let $g$ and $g_{0}$ be smooth metrics on $B$ . We pro…

Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature

Liam Mazurowski, Xin Zhou

Assume $h$ is a positive function on the unit three-sphere which satisfies the pinching condition $h < h_{0} \approx 0.547$ . We prove the existence of at least two embedded two-spher…

On the topology of manifolds with positive intermediate curvature

Liam Mazurowski, Tongrui Wang, Xuan Yao

We formulate a conjecture relating the topology of a manifold's universal cover with the existence of metrics with positive $m$ -intermediate curvature. We prove the result for mani…

Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds

Liam Mazurowski, Jintian Zhu

We prove the existence of compact surfaces with prescribed constant mean curvature in asymptotically flat and asymptotically hyperbolic manifolds. More precisely, let $(M^{3}, g)$ be…

Euclidean Domains with Nearly Maximal Yamabe Quotient

Liam Mazurowski, Xuan Yao

Let $Ω$ be a smooth, bounded domain in $R^{3}$ with connected boundary. It follows from work of Escobar that the Yamabe quotient of $Ω$ is at most the Yamabe quotient of a ba…

Mass, Conformal Capacity, and the Volumetric Penrose Inequality

Liam Mazurowski, Xuan Yao

Let $Ω$ be a smooth, bounded subset of $R^{3}$ diffeomorphic to a ball. Consider $M = R^{3} ∖ Ω$ equipped with an asymptotically flat metric $g = f^4 g_{\text…

An Alternative for Constant Mean Curvature Hypersurfaces

Liam Mazurowski, Xin Zhou

Let $M^{n + 1}$ be a closed manifold of dimension $3 \leq n + 1 \leq 7$ equipped with a generic Riemannian metric $g$ . Let $c$ be a positive number. We show that, either there exist infini…

Infinitely Many Half-Volume Constant Mean Curvature Hypersurfaces via Min-Max Theory

Liam Mazurowski, Xin Zhou

Let $(M^{n + 1}, g)$ be a closed Riemannian manifold of dimension $3 \leq n + 1 \leq 5$ . We show that, if the metric $g$ is generic or if the metric $g$ has positive Ricci curvature, then $…