arXiv subjects

Rafe Mazzeo

Publications and source records attributed to Rafe Mazzeo.

At least 19 recordsLinked to original sources

arXiv2026

Minimal Surfaces with Stratified Branching Sets

Inspired by the Taubes-Wu construction of $\mathcal{C}^{1,\alpha}$ two-valued harmonic functions by the use of symmetry, we construct minimal surfaces with stratified branching sets as graphs of $\mathcal{C}^{1,\alpha}$ two-valued functions. We give three constructions. The first is perturbative and produces branched minimal submanifolds in arbitrary codimension as two-valued graphs over the $n$-ball or, slightly more generally, over the product of $B^n$ with a torus $\mathbb T^N$, parametrized by boundary data which is required to be small in a suitable norm. The second uses barrier methods together with a reflection argument to produce branched stable minimal hypersurfaces, again as two-valued graphs over the unit $n$-ball or $B^n \times \mathbb T^N$, parametrized by boundary data which now can be large. Finally, using bifurcation theory, we produce compact minimal submanifolds with similarly stratified branching sets in an ambient space $S^n \times \mathbb R$ with a suitable (analytic) warped product metric. These examples give minimal submanifolds with novel frequency values and whose branching sets have non-trivial deeper strata. While the main constructions are fairly elementary, they rely on the use of precisely tailored (and somewhat non-standard) function spaces, combined with a regularity theory which provides full asymptotic expansions around the branching sets.

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arXiv2026

ALG gravitational instantons and Hitchin moduli spaces, I: Torelli parameters

This is the first of two papers which together prove that the $12$-parameter family of parabolic $SU(2)$-Hitchin moduli spaces on the four-punctured sphere are all ALG gravitational instantons of type D4, and hence are asymptotic to $(\mathbb{C} \times T^2_\tau)/\mathbb{Z}_2$ at infinity. The elliptic modulus $\tau$ is determined by the cross-ratio of the four points. In this first paper, we consider each Hitchin moduli space corresponding to an allowable set of parabolic data and compute its Torelli parameters. There is a $12$-parameter family of Hitchin moduli spaces corresponding to different parabolic data, and we show that these realize all possible allowable Torelli parameters. In the companion paper, we we will show there that all of the Hitchin moduli spaces studied here are indeed ALG of type $D_4$, and consequently that every ALG-$D_4$ gravitational instanton can be realized as a Hitchin moduli space. Altogether, this will give the first verification of any case of the Modularity Conjecture: that all ALG gravitational instantons with tangent cone $\mathbb{C}/\mathbb{Z}_2$ can be realized as Hitchin moduli spaces with their natural associated $L^2$ metrics.

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arXiv2025

Tian--Yau metrics: Fredholm theory, Hodge cohomology and moduli spaces

We study the natural geometric elliptic operators on a class of complete Riemannian manifolds which include the 4-dimensional ALH* gravitational instantons and their higher dimensional Calabi-Yau analogues asymptotic to the model Calabi Ansatz metrics. Some of these were initially constructed by Tian and Yau, later by Hein and most recently by Y. Chen, and we call these Tian-Yau spaces. They have played an important role in the degeneration theory of K3 metrics, cf. Hein-Sun-Viaclosky-Zhang and Sun-Zhang, in the increasingly refined classification of gravitational instantons (cf. Collins-Jacob-Lin, Lee-Lin), and other areas. We show that these elliptic operators can be analyzed using the $\mathbf{a}$-pseudodifferential calculus of Grieser and Hunsicker, and use this to determine the space of $L^2$ harmonic forms in the four-dimensional setting, as well as the refined asymptotic regularity and local deformation theory of ALH* structures.

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arXiv2025

Linear and nonlinear stability for the Bach flow, I

In this paper we prove the linear stability of a gauge-modified version of the Bach flow on any complete manifold (M, h) of constant curvature. This involves some intricate calculations to obtain spectral bounds, and in particular introduces a higher order generalization of the well-known Koiso identity. We also prove nonlinear stability for the Bach flow if (M, h) is hyperbolic space, and more generally any Poincar\'e-Einstein space sufficiently close to h. In the forthcoming Part II of this project, we study the nonlinear stability question if M is either compact or else noncompact and flat, since those cases require different considerations involving a center manifold.

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arXiv2024

Double b-fibrations and desingularization of the X-ray transform on manifolds with strictly convex boundary

We study the mapping properties of the X-ray transform and its adjoint on spaces of conormal functions on Riemannian manifolds with strictly convex boundary. After desingularizing the double fibration, and expressing the X-ray transform and its adjoint using b-fibrations operations, we employ tools related to Melrose's Pushforward Theorem to describe the mapping properties of these operators on various classes of polyhomogeneous functions, with special focus to computing how leading order coefficients are transformed. The appendix explains that a naïve use of the Pushforward Theorem leads to a suboptimal result with non-sharp index sets. Our improved results are obtained by closely inspecting Mellin functions which arise in the process, showing that certain coefficients vanish. This recovers some sharp results known by other methods. A number of consequences for the mapping properties of the X-ray transform and its normal operator(s) follow.

math.AP↗

arXiv2024

Convergence stability for Ricci flow on manifolds with bounded geometry

We prove that the Ricci flow for complete metrics with bounded geometry depends continuously on initial conditions for finite time with no loss of regularity. This relies on our recent work where sectoriality for the generator of the Ricci-DeTurck flow is proved. We use this to prove that for initial metrics sufficiently close in Hölder norm to a rotationally symmetric asymptotically hyperbolic metric and satisfying a simple curvature condition, but a priori distant from the hyperbolic metric, Ricci flow converges to the hyperbolic metric.

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arXiv2024

Geometric inverse problems on gas giants

On gas giant planets the speed of sound is isotropic and goes to zero at the surface. Geometrically, this corresponds to a Riemannian manifold whose metric tensor has a conformal blow-up near the boundary. The blow-up is tamer than in asymptotically hyperbolic geometry: the boundary is at a finite distance. We study the differential geometry of such manifolds, especially the asymptotic behavior of geodesics near the boundary. We relate the geometry to the propagation of singularities of a hydrodynamic PDE and we give the basic properties of the Laplace--Beltrami operator. We solve two inverse problems, showing that the interior structure of a gas giant is uniquely determined by different types of boundary data.

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arXiv2024

Solving the Scattering Problem for Open Wave-Guide Networks, III: Radiation Conditions and Uniqueness

This paper continues the analysis of the scattering problem for a network of open wave-guides started in [arXiv:2302.04353, arXiv:2310.05816]. In this part we present explicit, physically motivated radiation conditions that ensure uniqueness of the solution to the scattering problem. These conditions stem from a 2000 paper of A. Vasy on 3-body Schrodinger operators; we discuss closely related conditions from a 1994 paper of H. Isozaki. Vasy's paper also proves the existence of the limiting absorption resolvents, and that the limiting solutions satisfy the radiation conditions. The statements of these results require a calculus of pseudodifferential operators, called the 3-body scattering calculus, which is briefly introduced here. We show that the solutions to the model problem obtained in arXiv:2302.04353 satisfy these radiation conditions, which makes it possible to prove uniqueness, and therefore existence, for the system of Fredholm integral equations introduced in that paper.

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arXiv2023

An index theorem for Z/2-harmonic spinors branching along a graph

We prove an index formula for the Dirac operator acting on two-valued spinors on a $3$-manifold $M$ which branch along a smoothly embedded graph $\Sigma \subset M$, and with respect to a boundary condition along $\Sigma$ inspired by an instance of this setting related to the deformation theory of $\mathbb Z_2$-harmonic spinors. When $\Sigma$ is a smooth embedded curve, this index vanishes; this was proved earlier by one of us, but the proof here is different and extends to the more general setting where $\Sigma$ also has vertices. We focus primarily on the Dirac operator itself, but also show how our results apply to more general twisted Dirac operators and to the closely related $\mathbb Z_2$ harmonic $1$-forms.

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arXiv2023

New examples of Z/2 harmonic 1-forms and their deformations

We collect a number of elementary constructions of $\Z_2$ harmonic $1$-forms, and of families of these objects. These examples show that the branching set $\Sigma$ of a $\Z_2$ harmonic 1-form may exhibit the following features: i) $\Sigma$ may be a non-trivial link; ii) $\Sigma$ may be a multiple cover; iii) $\Sigma$ may be immersed, and appear as a limit of smoothly embedded branching loci; iv) there are families of $\Z_2$ harmonic $1$-forms whose branching sets $\Sigma$ have tangent cones filling out a positive dimensional space, even modulo isometries. We show that Features i) and ii) occur already in dimension three, while the remaining ones appear at least in dimension four and higher.

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arXiv2023

The Algebraic and Analytic Compactifications of the Hitchin Moduli Space

Following the work of Mazzeo-Swoboda-Weiss-Witt and Mochizuki, there is a map $\overline{\Xi}$ between the algebraic compactification of the Dolbeault moduli space of $\mathsf{SL}(2,\mathbb{C})$ Higgs bundles on a smooth projective curve coming from the $\mathbb{C}^\ast$ action, and the analytic compactification of Hitchin's moduli space of solutions to the $\mathsf{SU}(2)$ self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ``limiting configurations''. This map extends the classical Kobayashi-Hitchin correspondence. The main result of this paper is that $\overline{\Xi}$ fails to be continuous at the boundary over a certain subset of the discriminant locus of the Hitchin fibration. This suggests the possibility of a third, refined compactification which dominates both.

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arXiv2022

Wellposedness of nonlinear flows on manifolds of bounded geometry

We present simple conditions which ensure that a strongly elliptic operator $L$ generates an analytic semigroup on Hölder spaces on an arbitrary complete manifold of bounded geometry. This is done by establishing the equivalent property that $L$ is "sectorial", a condition that specifies the decay of the resolvent $(λI - L)^{-1}$ as $λ$ diverges from the Hölder spectrum of $L$. As one step, we prove existence of this resolvent if $λ$ is sufficiently large, and on this general class of manifolds, use a geometric microlocal version of the semiclassical pseudodifferential calculus. The properties of $L$ and $e^{-tL}$ we obtain can then be used to prove wellposedness of a wide class of nonlinear flows. We illustrate this by proving wellposedness on Hölder spaces of the flow associated to the ambient obstruction tensor on complete manifolds of bounded geometry.

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arXiv2021

Asymptotic Geometry of the Moduli Space of Parabolic $SL(2,\mathbb{C})$-Higgs Bundles

Given a generic stable strongly parabolic $SL(2,\mathbb{C})$-Higgs bundle $(\mathcal{E}, φ)$, we describe the family of harmonic metrics $h_t$ for the ray of Higgs bundles $(\mathcal{E}, t φ)$ for $t\gg0$ by perturbing from an explicitly constructed family of approximate solutions $h_t^{\mathrm{app}}$. We then describe the natural hyperKähler metric on $\mathcal{M}$ by comparing it to a simpler "semi-flat" hyperKähler metric. We prove that $g_{L^2} - g_{\mathrm{sf}} = O(\mathrm{e}^{-γt})$ along a generic ray, proving a version of Gaiotto-Moore-Neitzke's conjecture. Our results extend to weakly parabolic $SL(2,\mathbb{C})$-Higgs bundles as well. In the case of the four-puncture sphere, we describe the moduli space and metric more explicitly. In this case, we prove that the hyperkähler metric is ALG and show that the rate of exponential decay is the conjectured optimal one, $γ=4L$, where $L$ is the length of the shortest geodesic on the base curve measured in the singular flat metric $|\mathrm{det}\, φ|$.

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arXiv2021

Conical metrics on Riemann surfaces, II: spherical metrics

We continue our study, initiated in our earlier paper, of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with some or all of the cone angles greater than $2π$. Deformations are obstructed precisely when the number $2$ lies in the spectrum of the Friedrichs extension of the Laplacian. Our main result is that, in this case, it is possible to find a smooth local moduli space of solutions by allowing the cone points to split. This analytic fact reflects geometric constructions in papers by Mondello and Panov.

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arXiv2021

Manifolds with many Rarita-Schwinger fields

The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected Kähler-Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi-Yau manifolds of even complex dimension with more linearly independent Rarita-Schwinger fields than flat tori of the same dimension.

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arXiv2021

Short-time existence for the network flow

This paper contains a new proof of the short-time existence for the flow by curvature of a network of curves in the plane. Appearing initially in metallurgy and as a model for the evolution of grain boundaries, this flow was later treated by Brakke \cite{Br} using varifold methods. There is good reason to treat this problem by a direct PDE approach, but doing so requires one to deal with the singular nature of the PDE at the vertices of the network. This was handled in cases of increasing generality by Bronsard-Reitich \cite{BrRe}, Mantegazza-Novaga-Tortorelli \cite{MNT} and eventually, in the most general case of irregular networks by Ilmanen-Neves-Schulze \cite{INS}. Although the present paper proves a result similar to the one in \cite{INS}, the method here provides substantially more detailed information about how an irregular network `resolves' into a regular one. Either approach relies on the existence of self-similar expanding solutions found in \cite{MS}. As a precursor to and illustration of the main theorem, we also prove an unexpected regularity result for the mixed Cauchy-Dirichlet boundary problem for the linear heat equation on a manifold with boundary.

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arXiv2019

Ricci Flow and Volume Renormalizability

With respect to any special boundary defining function, a conformally compact asymptotically hyperbolic metric has an asymptotic expansion near its conformal infinity. If this expansion is even to a certain order and satisfies one extra condition, then it is possible to define its renormalized volume and show that it is independent of choices that preserve this evenness structure. We prove that such expansions are preserved under normalized Ricci flow. We also study the variation of curvature functionals in this setting, and as one application, obtain the variation formula $$ \frac{\rm d}{{\rm d}t} {\rm RenV}\big(M^n, g(t)\big) = -\mathop{\vphantom{T}}^R \! \! \! \int_{M^n} (S(g(t))+n(n-1)) {\rm d}V_{g(t)},$$ where $S(g(t))$ is the scalar curvature for the evolving metric $g(t)$, and $\mathop{\vphantom{T}}^R \! \! \! \int (\cdot) {\rm d}V_g$ is Riesz renormalization. This extends our earlier work to a broader class of metrics.

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arXiv2019

Asymptotic Geometry of the Hitchin Metric

We study the asymptotics of the natural $L^2$ metric on the Hitchin moduli space with group $G = \mathrm{SU}(2)$. Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore \cite{gmn13}, is that on the regular part of the Hitchin system, this metric is well-approximated by the semiflat metric from \cite{gmn13}. We prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion, and hence that the the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay. Very recent work by Dumas and Neitzke indicates that the convergence rate for the metric is exponential, at least in certain directions.

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