Joint Japan/US Collaborative Workshop on Geometric Analysis II

September 12–15, 2025
Kyoto University, Kyoto, Japan

This workshop is the sequel to the previous event held at Stanford University in August 2023.

Invited Speakers

Venue

The workshop takes place in the Lecture Room 127 of the Department of Mathematics (Science Building No. 3), Kyoto University.

Schedule

Friday, September 12

Saturday, September 13

Sunday, September 14

Monday, September 15

Titles & Abstracts

Laura Fredrickson — The asymptotics of hyperkahler metrics from Gaiotto–Moore–Neitzke's conjecture
Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmüller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperkähler metric, a rich and rigid geometric structure. An intricate conjectural description of its asymptotic structure appears in the work of Gaiotto–Moore–Neitzke.  I will discuss some recent and less-recent results using tools coming out of geometric analysis which are well-suited for verifying these extremely delicate conjectures. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these conjectures.
Matthew Gursky — Some rigidity results for asymptotically hyperbolic Einstein metrics
In this talk I will describe some gap estimates for the renormalized volume of ‘even’ and self-dual AHE metrics in dimension four. Even AHE metrics naturally arise from a non-local variational problem, and there is an interesting parallel between uniqueness questions for Einstein metrics on $S^4$ and even AHE metrics on the ball. I will also describe a connection to the renormalized area of minimal surfaces. This is joint work with S. McKeown and A. Tyrrell.
Eiji Inoue — An algebro-geometric entropy formula for MMP with scaling
In his celebrated work, Perelman introduced two functionals ($F$ and $W$) which are monotone along Ricci flow. Consider the flow starting from a Kähler metric. Then the cohomology class moves linearly $L_t = L+tK_X$ and the flow runs as long as this class is positive. When $X$ is not a minimal model, this class becomes not positive at finite time and there appears an algebraic curve whose area vanishes. Then one can construct a morphism of algebraic varieties $X \to Y$ which smashes the curve to a point. The construction is purely algebro-geometric and is known as the first step of MMP with scaling. We introduce a purely algebro-geometric invariants of L which behave monotonically along this process. The invariants are deeply related to Perelman's entropy, Sun–Zhang's weighted volume and K-stability.
Demetre Kazaras — A non-optimal Penrose Inequality
To test the Weak Cosmic Censorship Hypothesis, Penrose proposed an inequality between the ADM mass $m$ of a spacetime and the cross-sectional area $A$ of its event horizon, which reads $16\pi m^2\geq A$. In this talk, we work with asymptotically flat initial data sets satisfying the dominant energy condition and show $C m^2>A_h$, where $A_h$ is the minimal area required to enclose an outermost apparent horizon and the (non-optimal) constant $C$ is less than $10^{18}$. The proof combines Schoen–Yau's analysis of the Jang equation, the harmonic function approach to the Positive Mass Theorem, and Dong–Song's stability argument. This is joint work with Brian Allen, Edward Bryden, and Marcus Khuri.
Daisuke Kishimoto — Morse inequalities for noncompact manifolds
I will talk about joint work with Tsuyoshi Kato and Mitsunobu Tsutaya, in which we establish Morse inequalities for an infinite Galois covering over a closed manifold, where inequalities are expressed by a rough configuration of critical points of a Morse function and $L^2$-Betti numbers. As a corollary, we get the mean value version of Morse inequalities and a new property of $L^2$-Betti numbers.
Keita Kunikawa — Index estimate by first Betti number of minimal hypersurfaces in compact symmetric spaces
I will show that the Morse index of an unstable closed minimal hypersurface in a compact semi-simple Riemannian symmetric space is bounded below by a dimensional constant multiple of the first Betti number of the hypersurface. The proof is inspired by the methods of Savo and Ambrozio–Carlotto–Sharp, whose approach is based on an “averaging method” using isometric immersions into Euclidean space. A key observation in our proof is that this method can be naturally extended using isometric immersions into compact semi-simple Riemannian symmetric spaces. This talk is based on a joint work with Toru Kajigaya.
Yoshihiko Matsumoto — Harmonic maps from the product of the hyperbolic planes to the hyperbolic space
I will discuss an existence result for the asymptotic Dirichlet problem for harmonic maps from the product of the hyperbolic planes to the hyperbolic space, where the Dirichlet data is given on the distiguished boundary (the product of the circles at infinity). This is based on joint work with Kazuo Akutagawa.
Rafe Mazzeo — Gravitational Instantons: ALH* spaces
After giving an overview of the emerging classification and structure theory of general gravitational instantons, i.e., complete hyperKähler 4-manifolds, I will discuss recent work with Xuwen Zhu concerning the somewhat ‘exotic’ class of ALH* instantons, with results on their Hodge theory, asymptotic regularity and deformation theory.
Paul Minter — Stationary integral varifolds near multiplicity 2 planes
A central question in the regularity theory of minimal surfaces is to better understand the behaviour of a convergent sequence of minimal surfaces when multiplicity occurs in the limit. In such situations we only know a very weak, measure-theoretic, sense of convergence holds, and structural properties of the sequence may be lost in the limit. A key situation in which this happens is when rescaling a fixed minimal surface about a branch point singularity; here, one can get a tangent cone which is a plane of (integer) multiplicity at least 2. Such compactness questions are also relevant in understanding moduli spaces in gauge theory.
I will discuss work which investigates this question in the simplest situation where the minimal surfaces in question are close to a plane of multiplicity 2. Here, we are able to give a structural condition which guarantees that the nearby minimal surface is in fact represented by the graph of a Lipschitz 2-valued function, with a priori estimates establishing, for instance, uniqueness of tangent cones locally. This can be viewed as a type of Allard regularity theorem for multiplicity 2 planes. This is based on joint work with Spencer Becker-Kahn and Neshan Wickramasekera.
Tatsuya Miura — A direction energy approach to elastica and elastic flow
The elastic energy, consisting of the bending energy and the length of curves, has critical points known as elastica, classical models of elastic rods studied since Euler, and generates a gradient flow called the elastic flow, a fundamental example of higher-order geometric flows. In this talk I will present the direction energy method, which augments the length integrand with a null Lagrangian, thereby yielding the same critical points and gradient flow. This modification provides a nontrivial finite energy for infinite-length curves, which unlocks a variational approach to infinite-length elastica and elastic flow. I will discuss applications to uniqueness results for elastica and to global properties of the elastic flow, including global well-posedness and sharp thresholds for embeddedness and graphicality. Based on joint works with Glen Wheeler (University of Wollongong) and Fabian Rupp (University of Vienna).
Makoto Nakamura — Global solutions for semi-linear Klein–Gordon equations in Friedmann–Lemaître–Robertson–Walker spacetimes
The Cauchy problem for the semi-linear Klein–Gordon equation is considered in Friedmann–Lemaître–Robertson–Walker spacetimes. The global well-posedness of the Cauchy problem is considered in Sobolev spaces. The spatial expansion yields a dissipative effect to the Klein–Gordon equation, which is expressed by the property of dissipative wave equations in the energy estimate. This property is used to solve the Cauchy problem. The non-existence of global solutions is also considered if time allows.
Yasufumi Nitta — Extremal Kähler metrics and Mabuchi solitons on Fano manifolds
In this talk, we concern the relation between two kinds of canonical Kähler metrics on Fano manifolds, the Calabi's extremal Kähler metrics and the Mabuchi solitons. These are both generalizations of the concept of Kähler-Einstein metrics. It is known that the existence of Mabuchi solitons implies that of extremal Kähler metrics representing the first Chern class. It is also known that the converse is true for Fano manifolds of dimension up to two. Based on the above, we present examples of Fano manifolds in all dimensions greater than two which admit extremal Kähler metrics in every Kähler class, but do not admit Mabuchi solitons. A key concept is a holomorphic invariant of Fano manifolds called the Mabuchi constant, which is smaller than 1 for Fano manifolds admitting a Mabuchi soliton. Furthermore, we show that for Fano manifolds whose Mabuchi constants are smaller than 1, the existence of extremal Kähler metrics representing the first Chern class implies that of Mabuchi solitons. If time allows, we will explain the related work by Hisamoto and Nakamura. This is partly joint work with Shunsuke Saito, and also partly joint work with Vestislav Apostolov and Abdellah Lahdili.
Hirofumi Sasahira — Seiberg–Witten Floer spectra for families of 3-manifolds
The Seiberg–Witten Floer spetrum is an invariant for 3-manifolds which is defined as an $S^1$-equivariant spectrum. It is known that its $S^1$-equivariant homology is isomorphic to the monopole Floer homology. The Seiberg–Witten Floer spectra have been producing interesting topological applications. In this talk, we introduce Seiberg-Witten Floer spectra for families of 3-manifolds and discuss their applications. This talk is based on joint work with Stoffregen and another project with Konno.
Masaki Taniguchi — Gauge theory on non-compact 4-manifolds and its applications to geometry
Yang–Mills and Seiberg–Witten gauge theories have revealed deep connections with the geometry of 4-manifolds. These theories are based on systems of first-order nonlinear PDEs derived from the geometric structure of 4-manifolds. In this talk, we first review the historical developments of gauge theory in 4-dimensional geometry. We then focus on the developments of these theories to non-compact 4-manifolds. In particular, we discuss applications to the non-existence of positive scalar curvature metrics and to obstructions for certain symplectic structures on compact 4-manifolds within specific classes.
Sumio Yamada — Einstein spacetime, harmonic maps and singularities
In 1923, Bach and Weyl constructed a 4dimentional static Einstein spacetime by juxtaposing two copies of Schwarzschild metrics. The construction is natural, but the resulting spacetime has a singularity set, which is physically unnatural. In a collaborative project with Marcus Khuri and Gilbert Winstein over the last decade, we have generalized the Bach-Weyl construction to 5-dimensional Einstein spacetimes with axial symmetries, using harmonic maps (also called non-linear sigma model) and constructed a new set of vacuum stationary Einstein solutions with and without singularities. We will discuss the connection between topology of the spaces and existence of singularities, and pose some open questions in the geometry of Ricci flat metrics.

Organizers

Local Organizers