Qiuye Jia
For a time dependent Schrödinger equation, the scattering map is the map sending the asymptotic profile of solution as $t\to-\infty$ to its asymptotic profile as $t\to+\infty$. In this paper we show that, for certain class of metrics, the scattering maps associated to two Schrödinger operators with two time dependent metrics only differ by a compact operator if and only if these two metrics are related by a pull-back of a diffeomorphism.
Allen Weitsman
Graphs of solutions to the minimal surface equation over simply connected domains with boundary values 0 can have at most exponential growth.
Yue Xin, Yan Li, Bingzhe Hou
In this paper, we study the right invariant metric $d_{H^{\infty}}$ on the analytic automorphism group $\rm{Aut}(\mathbb{D})$ of the unit open disk $\mathbb{D}$ induced by maximal modulus, that is, $d_{H^{\infty}}(\varphi, ψ)=\sup_{z\in\mathbb{D}}|\varphi(z)-ψ(z)|$ for any $\varphi, ψ\in \rm{Aut}(\mathbb{D})$. We give the explicit formula of the right invariant metric $d_{H^{\infty}}$ and characterize the almost regular Finsler geometric structure of $(\rm{Aut}(\mathbb{D}), d_{H^{\infty}})$.
Huy The Nguyen, Shengwen Wang
We study solutions to the self-dual Abelian Yang--Mills--Higgs (YMH) equations in the singular limit $\e \to 0 $, where the associated self-dual Ginzburg--Landau type energy \begin{align*} E_\e\begin{pmatrix}u\\ A\end{pmatrix} = \int_M \left( |\nabla^A u|^2 + \e^2 |F_A|^2 + \frac{(1 - |u|^2)^2}{4\e^2} \right) \mathrm{dvol}_g \end{align*} exhibits concentration along codimension-two sets. Using techniques inspired by Allard's regularity theory, we construct approximate solutions concentrating near a minimal submanifold and analyse their perturbations via a linearised operator projected orthogonally to gauge and translational zero modes. By working in Fermi coordinates and enforcing Coulomb gauge conditions, we derive uniform Lipschitz and curvature estimates for the solutions and obtain Hölder regularity for the scalar and connection components. These results establish a geometric framework for understanding vortex sheet formation and provide a regularity theory for the limiting defect set in the context of Abelian gauge theories.
Andreas Mueller
Many mechanical systems exhibit changes in their kinematic topology altering the mobility. Ideal contact is the best known cause, but also stiction and controlled locking of parts of a mechanism lead to topology changes. The latter is becoming an important issue in human-machine interaction. Anticipating the dynamic behavior of variable topology mechanisms requires solving a non-smooth dynamic problem. The core challenge is a physically meaningful transition condition at the topology switching events. Such a condition is presented in this paper. Two versions are reported, one using projected motion equations in terms of redundant coordinates, and another one using the Voronets equations in terms of minimal coordinates. Their computational properties are discussed. Results are shown for joint locking of a planar 3R mechanisms and a 6DOF industrial manipulator.
Leonardo F. Cavenaghi, Lino Grama, Ludmil Katzarkov, Pedro Antonio Muniz Martins
This paper investigates the geometric and cohomological properties of non-Kähler SYZ mirror symmetry for dual torus fibrations over solvmanifolds in the sense of Lau, Tseng and Yau. We are mainly concerned with three questions: \textbf{(a)} How the Lau-Tseng-Yau notion of non-Kähler SYZ is related to the mapping of supersymmetric branes between symplectic and complex sides; \textbf{(b)} Finding explicit non-Kähler SYZ mirror pairs determined purely by Lie-theoretic data; \textbf{(c)} better understand the cohomological correspondence in the Lau-Tseng-Yau framework (given by a Fourier-Mukai transform), especially concerning the role of Tseng-Yau cohomology. We prove that the Fourier-Mukai transform introduced by Lau-Tseng-Yau exchanges type-A supersymmetric cycles, which are given by special Lagrangian sections equipped with flat $\mathrm{U}(1)$ connections, with type-B cycles, corresponding to line bundles whose connections satisfy the deformed Hermitian-Yang-Mills (dHYM) equation. We provide pure Lie-theoretic criteria for the existence of non-Kähler SYZ mirror pairs whose base manifolds are solvmanifolds. Applying these criteria, we construct new explicit families of mirror pairs from almost abelian and generalized Heisenberg Lie groups, and provide a complete classification of such pairs arising from nilpotent Lie groups. To contextualize the role of the Tseng-Yau cohomology, we link it to noncommutative geometry. We introduce the Tseng-Yau and Bott-Chern mirror bicomplexes. We show that (some of) their enclosed cohomologies reduce to the primitive Tseng-Yau and Bott-Chern cohomologies and that for basic forms they are isomorphic under the Fourier-Mukai transform. As a last contribution, we discuss how to explicitly compute the Tseng-Yau and the Bott-Chern cohomology for the non-Kähler SYZ mirror pairs constructed here.
L. V. Bogdanov
We construct Orlov-Schulman symmetries for the self-dual conformal structure (SDCS) hierarchy. We provide an explicit proof of compatibility of additional symmetries with the basic Lax-Sato flows of the hierarchy, and consider several simple examples, including Galilean transformations and scalings. We also present a picture of the Orlov-Schulman symmetries in terms of a dressing scheme based on the Riemann-Hilbert problem.
Hanzhang Yin
In this paper, we study several types of geometric problems related to the Ricci curvature on noncompact complex manifolds, such as the existence of Kähler-Einstein metrics on complete Kähler manifolds with negative Ricci curvature, which can be seen as an improvement of the main theorem in Cheng-Yau [4]; the existence of canonical Hermitian metrics with prescribed Ricci curvature on complete Hermitian manifolds, which can be regarded as noncompact versions of the Gauduchon conjecture on certain complete complex surfaces. Our method can also be used to construct Hesse-Einstein metrics in affine differential geometry.
David Baraglia, Pedram Hekmati
We compute the Floer homology and Seiberg-Witten Floer homotopy type of Seifert rational homology $3$-spheres which fiber over $\mathbb{RP}^2$. We show that they are all $L$-spaces and their Floer homotopy type is a suspension of $S^0$. Additionally, we compute the Ozsváth-Szabó $d$-invariants, or equivalently the Seiberg-Witten $δ$-invariants for such $3$-manifolds. This is done by computing the eta invariant of spin$^c$-Dirac operators associated to spin$^c$-connections covering the adiabatic connection, a certain metric connection distinct from the Levi-Civita connection. It turns out that this eta invariant involves a contribution given by the eta invariant of an orbifold pin$^c$-connection on the orbifold base of the Seifert fibration, which we also compute.
Tamás Darvas
We survey selected developments in the metric geometry of the space of Kähler metrics, emphasizing results from the past decade, highlighting open problems along the way.
Emily B. Dryden, Carolyn Gordon, Javier Moreno, Julie Rowlett, Carlos Villegas-Blas
The extent to which the geometry of an object is determined by some associated spectral data is a longstanding problem. We investigate this problem in the context of the Steklov spectrum, focusing on convex polygons. We prove that almost all triangles are uniquely determined by their Steklov spectra within the class of all triangles; further results depending on the types of angles in the triangles are given. We examine three special classes of convex quadrilaterals--rectangles, parallelograms, and kites--and obtain results ranging from unique spectral determination to determination up to three possibilities. For regular $n$-gons, we are again able to prove spectral determination within certain classes of polygons. More generally, we investigate the extent to which the Steklov spectrum distinguishes convex polygons from simply-connected domains with smooth boundary; that is, does the Steklov spectrum detect corners? We prove that triangles and quadrilaterals are spectrally distinguished from such smoothly bounded domains; moreover, we show that having the same Steklov spectrum as such a domain imposes substantial restrictions on the edge lengths of higher-order $n$-gons. Throughout, our main tool is the characteristic polynomial developed in works by Stanislav Krymski, Michael Levitin, Leonid Parnovski, Iosif Polterovich, and David A. Sher.
Jorge Herbert Soares de Lira, Rafael Rocha de Farias
In this paper we prove existence and classification results for translating solitons defined as initial conditions for higher order mean curvature flows that are invariant by translations in warped product manifolds $\mathbb{P}\times_χ\mathbb{R}$. Here, $\mathbb P$ is a Cartan-Hadamard manifold endowed with a rotationally symmetric metric and $χ$ is a radial function defined in $\mathbb{P}$. In this setting, the higher order mean curvature flow is, up to a change of time parameter, given by translations along the factor $\mathbb{R}$ in the warped product. This setting encompasses the cases of translating solitons in $\mathbb{R}^{n+1}$, $\mathbb{H}^n \times \mathbb{R}$ and $\mathbb{H}^{n+1}$ studied in recent papers. In particular we prove the existence of families of bowl-type and catenoid-type translating solitons under mild assumptions about the curvature of the warped product. We also describe the asymptotic behavior for those solitons in terms of the geometry at infinity of $\mathbb{P}$. Our assumptions about the ambient metric allow us to control the higher order mean curvature of cylinders and to use them as barriers.
Douglas Stryker
We give a proof that every complete two-sided stable minimal surface in $\mathbb{R}^3$ is flat using the index theory for Dirac operators on twisted spinor bundles.
Marcos M. Alexandrino, Benigno O. Alves, Patricia Marcal
We investigate singular Finsler foliations (SFFs) on a manifold equipped with an $(α,β)$-metric. To be precise, we verify that any SFF of an $(α,β)$-space is, under some hypotheses on the metric, a singular Riemannian foliation (SRF). This gives a partial answer to the general question "under which conditions a SFF is a SRF with respect to some Riemannian metric". Moreover, we extend the proof of Molino's conjecture to SFFs whenever they are also a SRFs. Finally, we prove equifocality of the regular leaves for a SFF under the same condition.
S. Brendle, Y. Wang
We describe how the spacetime positive energy theorem in dimension $n \geq 4$ follows from our recent work on the Riemannian version of the positive mass theorem. Our proof builds on the fundamental work of Schoen and Yau and the remarkable work of Eichmair, and uses the Jang equation with a capillary term. We also use the shielding principle from the work of Lesourd-Unger-Yau.
Amanda Maria Petcu
A conjecture of Simon Donaldson is that on a compact $4$-manifold $X^4$ one can flow from a hypersymplectic structure to a hyperkähler structure while remaining in the same cohomology class. To this end the hypersymplectic flow was introduced by Fine-Yao. In this paper the notion of a positive triple on $X^4$ is used to describe a hypersymplectic and hyperkähler structure. Given a closed positive triple one can define either a closed $G_2$ structure or a coclosed $G_2$ structure on $\mathbb{T}^3 \times X^4$. The coclosed $G_2$ structure is evolved under the modified $G_2$-Laplacian coflow. The coflow descends to a flow of the positive triple on $X^4$, which is again the Fine-Yao hypersymplectic flow.
Zhufeng Yao
Let $Γ\subset \mathsf{PGL}(d,\mathbb{R})$ be an irreducible projective Anosov subgroup and let $Λ^1(Γ)$ be its projective limit set. Viewing $Λ^1(Γ)$ as an analogue of a self-affine set, we investigate the Hausdorff dimension of $Λ^1(Γ)$ under specific assumptions regarding its affine complexity: 1. If $Λ^1(Γ)$ is of full Hausdorff dimension, then $d= 2$ and $Γ$ is a cocompact lattice. 2. If $d = 3$ and $Γ$ is the image of a closed surface group under an irreducible Anosov representation, then $Λ^1(Γ)$ never has Hausdorff dimension $1$ unless the representation is Hitchin. 3. If the limit set $Λ^1(Γ)$ exhibits a partial quasi-self-similarity (in the sense of Falconer~\cite{falconerselfsimilar1}) -- which can be implied by the ``regular distortion property'' of $Γ$ -- then the Hausdorff dimension of $Λ^1(Γ)$ equals the critical exponent of the first simple root. An application of this result is the computation of the Hausdorff dimension of the limit set for arbitrary $Θ$-positive representations of convex cocompact Fuchsian groups.
Marco Gallo, Luigi Vezzoni
We establish a general result ensuring a $C^1$ a priori bound for smooth curves of Hermitian metrics. As a main application, we obtain a new regularity result for Hermitian curvature flows, and in particular for the second Chern-Ricci flow.
Andrew Gracyk
We develop interconnections between the complex normalizing flow for data drawn from Borel probability measures on the twofold realification of the complex manifold and the Kähler-Ricci flow. The complex normalizing flow relates the initial and target realified densities under the complex change of variables, necessitating the log determinant of the Wirtinger Jacobian. The Ricci curvature of a Kähler manifold is the second order mixed Wirtinger partial derivative of the log of the local density of the volume form. Therefore, we reconcile these two facts by drawing forth the connection that the log determinant used in the complex normalizing flow matches the Ricci curvature term under differentiation and conditions. The log density under the normalizing flow is kindred to a spatial Fisher information metric under a holomorphic pullback and a Bayesian perspective to the parameter, thus under the continuum limit the log likelihood matches a Fisher metric, recovering the Kähler-Ricci flow up to expectation. Using this framework, we establish other relevant results, attempting to bridge the statistical and ordinary behaviors of the complex normalizing flow to the geometric features of the Kähler-Ricci flow.
Michael Eastwood, Thomas Leistner
We present a systematic prolongation procedure and its implementation for Killing two-tensors, especially in the locally symmetric case. We use the resulting machinery to elucidate the natural quadratic mapping from Killing fields to Killing two-tensors on irreducible locally symmetric spaces of compact type.