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Liam Mazurowski

Publications and source records attributed to Liam Mazurowski.

At least 19 recordsLinked to original sources
arXiv2026

A Poincare-Einstein Liouville Theorem and the Yamabe Invariant of Lens Spaces

Schoen conjectured that the Yamabe invariant of every lens space is achieved by the round metric. A proof of this conjecture is presented for all lens spaces, together with rigidity: equality is only achieved for the round class. The key step is to show that every $\mathbb Z_p$-invariant Yamabe-positive conformal class on $S^3$ admits a $\mathbb Z_p$-equivariant Poincaré-Einstein filling. The distance function from a fixed point of the filling then gives an invariant test function with Yamabe quotient at most $σ(S^3)$, which descends to the lens space. The proof of the main theorem was generated by AI (ChatGPT Astra, September 2026); this is mainly a learning note of the AI generated proof.

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arXiv2026

Scalar Curvature Lower Bounds and Convergence in Measure

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 it follow that $g$ also has non-negative scalar curvature? We answer Gromov's question in dimension three. We show that, without further assumptions, the metric $g$ need not have non-negative scalar curvature and in fact $g$ may be completely arbitrary. However, if one assumes in addition that the identity maps $(M,g_k)\to (M,g)$ are uniformly bi-Lipschitz, then indeed $g$ must have non-negative scalar curvature. Our method of proof can also be used to show that, under an almost Euclidean entropy condition, scalar curvature lower bounds will persist under convergence of the metrics together with their inverses in $L^p$ for suitably large $p$.

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arXiv2026

A Positive Mass Theorem for Continuous Metrics

Let $g$ be a continuous metric on $\mathbb R^3$ which is asymptotically flat in the sense that $\vert g_{ij}(x) - δ_{ij}\vert = O(\vert x\vert^{-τ})$ for some $τ> \frac{1}{2}$. Further assume that $g$ can be uniformly approximated on compact sets by smooth metrics with almost non-negative scalar curvature. For such a metric $g$, we define a synthetic ADM mass $m(g)$ using harmonic functions. The harmonic mass $m(g)$ coincides with the usual ADM mass whenever $g$ is smooth and decays rapidly enough that the latter is defined. The harmonic mass can also be computed as a limit of the $C^0$ local mass introduced by Burkhardt-Guim. Our main result is a positive mass theorem: the harmonic mass satisfies $m(g)\geq 0$ and if $m(g) = 0$ then $g$ is flat.

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arXiv2026

Scalar curvature under weak limits of manifolds

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 smooth, closed, Riemannian three manifolds. Assume that there are smooth, surjective, $λ_k$-Lipschitz maps $f_k\colon M_k \to M$ and that $\text{Vol}(M_k)\to \text{Vol}(M)$ and $λ_k\to 1$. Then if each $M_k$ has scalar curvature bounded below by $κ$ so does $M$. This result answers questions of Gromov, Sormani, Allen, and others. The proof relies on a delicate comparison between $μ$-bubbles in $M_k$ and $μ$-bubbles in $M$.

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arXiv2026

Quantification of $C^0$ Convergence in Dimension Three

We address Gromov's Quantification of $C^0$ Convergence Conjecture in dimension three. Let $B$ be the unit ball in $\mathbb R^3$. Let $g$ and $g_0$ be smooth metrics on $B$. We prove there are constants $C$ and $ε_0$ depending only on $g_0$ so that \[ \inf_{x\in B} R_g(x) \leq R_{g_0}(0) + C \|g-g_0\|_{C^0}^{1/2} \] provided $\|g-g_0\|_{C^0}\leq ε_0$. We also construct examples to show that the exponent $1/2$ is sharp. This explicitly quantifies the fact that scalar curvature lower bounds are preserved under $C^0$ convergence of metrics. When $g_0$ is merely $C^2$ we prove a related estimate with a slightly weaker rate, and when $g_0$ has rotational symmetry we prove a related estimate with a stronger linear rate. To prove these results, we use harmonic functions to define a local quantity that detects the scalar curvature. Then we use classical elliptic PDE estimates to show that this quantity is stable under $C^0$ perturbations of the metric. As a further application of this method, we give a partial answer to a question of Gromov on the preservation of scalar curvature lower bounds for metrics that are converging in measure.

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arXiv2025

Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature

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-spheres with prescribed mean curvature $h$. The same result holds for sign-changing functions $h$ satisfying $\vert h\vert < h_0$ under a mild assumption on the zero set.

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arXiv2025

On the topology of manifolds with positive intermediate curvature

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 manifolds of dimension $n\in\{3,4,5\}$ and for most choices of $m$ when $n=6$. As a corollary, we show that a closed, aspherical 6-manifold cannot admit a metric with positive $4$-intermediate curvature.

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arXiv2025

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

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 an asymptotically flat manifold with scalar curvature $R\ge 0$. Then, for each constant $c>0$, there exists a compact, almost-embedded, free boundary constant mean curvature surface $Σ\subset M$ with mean curvature $c$. Likewise, let $(M^3,g)$ be an asymptotically hyperbolic manifold with scalar curvature $R\ge -6$. Then, for each constant $c>2$, there exists a compact, almost-embedded, free boundary constant mean curvature surface $Σ\subset M$ with mean curvature $c$. The proof combines min-max theory with the following fact about inverse mean curvature flow which is of independent interest: for any $T$ the inverse mean curvature flow emerging out of a point $p$ far enough out in an asymptotically flat (or asymptotically hyperbolic) end will remain smooth for all times $t\in (-\infty,T]$.

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arXiv2025

Euclidean Domains with Nearly Maximal Yamabe Quotient

Let $Ω$ be a smooth, bounded domain in $\mathbb R^3$ with connected boundary. It follows from work of Escobar that the Yamabe quotient of $Ω$ is at most the Yamabe quotient of a ball, and equality holds if and only if $Ω$ is a ball. We show that if equality almost holds then the following things are true: (i)$Ω$ is diffeomorphic to a ball; (ii) There is a small number $ε> 0$ such that $B(x,r) \subset Ω\subset B(x,r(1+ε))$; (iii) After suitable scaling, $Ω$ is Gromov-Hausdorff close to the unit ball when considered as a metric space with its induced length metric. We also give a qualitative comparison between $Q$ and the coefficient of quasi-conformality studied in the theory of quasi-conformal maps.

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arXiv2024

Mass, Conformal Capacity, and the Volumetric Penrose Inequality

Let $Ω$ be a smooth, bounded subset of $\mathbb{R}^3$ diffeomorphic to a ball. Consider $M = \mathbb{R}^3 \setminus Ω$ equipped with an asymptotically flat metric $g = f^4 g_{\text{euc}}$, where $f\to 1$ at infinity. Assume that $g$ has non-negative scalar curvature and that $Σ= \partial M$ is a minimal 2-sphere in the $g$ metric. We prove a sharp inequality relating the ADM mass of $M$ with the conformal capacity of $Ω$. As a corollary, we deduce a sharp lower bound for the ADM mass of $M$ in terms of the Euclidean volume of $Ω$. We also prove a stability type result for this ``volumetric Penrose inequality.'' The proofs are based on a monotonicity formula holding along the level sets of a 3-harmonic function.

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arXiv2024

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

Let $(M^{n+1},g)$ be a closed Riemannian manifold of dimension $3\le n+1\le 5$. We show that, if the metric $g$ is generic or if the metric $g$ has positive Ricci curvature, then $M$ contains infinitely many geometrically distinct constant mean curvature hypersurfaces, each enclosing half the volume of $M$. As an essential part of the proof, we develop an Almgren-Pitts type min-max theory for certain non-local functionals of the general form $$Ω\mapsto \operatorname{Area}(\partial Ω) - \int_Ωh + f(\operatorname{Vol}(Ω)).$$

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arXiv2024

An Alternative for Constant Mean Curvature Hypersurfaces

Let $M^{n+1}$ be a closed manifold of dimension $3\le n+1\le 7$ equipped with a generic Riemannian metric $g$. Let $c$ be a positive number. We show that, either there exist infinitely many distinct closed hypersurfaces with constant mean curvature equal to $c$, or there exist infinitely many distinct closed hypersurfaces with constant mean curvature less than $c$ but enclosing half the volume of $M$.

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arXiv2024

On the stability of the Yamabe invariant of $S^3$

Let $g$ be a complete, asymptotically flat metric on $\mathbb{R}^3$ with vanishing scalar curvature. Moreover, assume that $(\mathbb{R}^3,g)$ supports a nearly Euclidean $L^2$ Sobolev inequality. We prove that $(\mathbb{R}^3,g)$ must be close to Euclidean space with respect to the $d_p$-distance defined by Lee-Naber-Neumayer. We then discuss some consequences for the stability of the Yamabe invariant of $S^3$. More precisely, we show that if such a manifold $(\mathbb{R}^3,g)$ carries a suitably normalized, positive solution to $Δ_g w + λw^5 = 0$ then $w$ must be close, in a certain sense, to a conformal factor that transforms Euclidean space into a round sphere.

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arXiv2024

Monotone Quantities for $p$-Harmonic functions and the Sharp $p$-Penrose inequality

Consider a complete asymptotically flat 3-manifold $M$ with non-negative scalar curvature and non-empty minimal boundary $Σ$. Fix a number $1 < p < 3$. We derive monotone quantities for $p$-harmonic functions on $M$ which become constant on Schwarzschild. These monotonicity formulas imply a sharp mass-capacity estimate relating the ADM mass of $M$ with the $p$-capacity of $Σ$ in $M$, which was first proved by Xiao using weak inverse mean curvature flow.

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arXiv2023

Curvature estimates for stable free boundary minimal hypersurfaces in locally wedge-shaped manifolds

In this paper, we consider locally wedge-shaped manifolds, which are Riemannian manifolds that are allowed to have both boundary and certain types of edges. We define and study the properties of free boundary minimal hypersurfaces inside locally wedge-shaped manifolds. In particular, we show a compactness theorem for free boundary minimal hypersurfaces with curvature and area bounds in a locally wedge-shaped manifold. Additionally, using Schoen-Simon-Yau's estimates, we also prove a Bernstein-type theorem indicating that, under certain conditions, a stable free boundary minimal hypersurface inside a Euclidean wedge must be a portion of a hyperplane. As our main application, we establish a curvature estimate for sufficiently regular free boundary minimal hypersurfaces in a locally wedge-shaped manifold with certain wedge angle assumptions. We expect this curvature estimate will be useful for establishing a min-max theory for the area functional in wedge-shaped spaces.

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arXiv2023

Min-max theory for free boundary minimal hypersurfaces in locally wedge-shaped manifolds

We develop a min-max theory for the area functional in the class of locally wedge-shaped manifolds. Roughly speaking, a locally wedge-shaped manifold is a Riemannian manifold that is allowed to have both boundary and certain types of edges. Fix a dimension $3 \le n+1 \le 6$. As our main theorem, we prove that every compact locally wedge-shaped manifold $M^{n+1}$ with acute wedge angles contains a locally wedge-shaped free boundary minimal hypersurface $Σ^n$ which is smooth in its interior and on its faces and is $C^{2,α}$ up to and including its edge. We can also handle the case of 90 degree wedge angles under an additional assumption.

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