arXiv research

Geodesics in non-Archimedean metrics are continuous.

Study finite-energy metrics over complex manifold degenerations.

We introduce different Finsler metrics on the space of smooth Kähler potentials that will induce a natural geometry on various finite energy classes $\mathcal E_{\tilde χ}(X,ω)$. Motivated by questions raised by R. Berman, V. Guedj and Y. Rubinstein, we characterize the underlying topology of these spaces in terms of c…

The study constructs universal invariants for non-Archimedean metrics on projective varieties.

New metric spaces for geodesic rays in cohomology classes.

Researchers confirm scalar-flatness for critical metrics in 5-9 dimensions.

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$, and $θ$ be a closed smooth real $(1,1)$-form representing a big and nef cohomology class. We introduce a metric $d_p, p\geq 1$, on the finite energy space $\mathcal{E}^p(X,θ)$, making it a complete geodesic metric space.

The space of Kähler metrics can, on the one hand, be approximated by subspaces of algebraic metrics, while, on the other hand, can be enlarged to finite-energy spaces arising in pluripotential theory. The latter spaces are realized as metric completions of Finsler structures on the space of Kähler metrics. The former s…

Synthetic approach to pluripotential theory measures finite energy.

New rigidity results for critical metrics of a quadratic curvature functional.

Assume $(X, ω)$ is a compact symplectic manifold with a Hamiltonian compact Lie group action and the zero in the Lie algebra is a regular value of the moment map $μ$. We prove that a finite energy symplectic vortex exponentially converges to (un)twisted sectors of the symplectic reduction at cylinder ends whose metrics…

The paper proves new rigidity results for critical metrics of quadratic curvature functionals.

Finite energy solutions of 4-harmonic and ES-4-harmonic maps are trivial.

A self-dual harmonic 2-form on a 4-dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form's zero set, the metric with the 2-form give a compatible almost complex structure and thus pseudo-holomorphic subvarieties. Such a subvariety is said to have finite energy when the …

Proves conditions for minimal surfaces in complex hyperbolic space.

We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for the infinitesimal generator. As examples we discuss Euclidean spaces, Riemannian…

The paper proves the existence of finite-energy solutions to a specific elliptic equation on Riemannian manifolds.

Study on finite entropy and energy in Kähler geometry.

Let $X$ be a compact Kähler manifold and $\a \in H^{1,1}(X,\R)$ a Kähler class. We study the metric completion of the space $\HH_\a$ of Kähler metrics in $\a$, when endowed with the Mabuchi $L^2$-metric $d$. Using recent ideas of Darvas, we show that the metric completion $(\overline{\HH}_\a,d)$ of $(\HH_\a,d)$ is a CA…

Study on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.

Let $X$ be a compact Kähler unibranch complex analytic space of pure dimension. Fix a big class $α$ with smooth representative $θ$ and a model potential $φ$ with positive mass. We define and the study non-pluripolar products of quasi-plurisubharmonic functions on $X$. We study the spaces $\mathcal{E}^p(X,θ;[φ])$ of fin…

Let $(X,ω)$ be a compact Kähler manifold and $\mathcal H$ the space of Kähler metrics cohomologous to $ω$. If a cscK metric exists in $\mathcal H$, we show that all finite energy minimizers of the extended K-energy are smooth cscK metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As a…

Suppose $(X,ω)$ is a compact Kähler manifold of dimension $n$, and $θ$ is closed $(1,1)$-form representing a big cohomology class. We introduce a metric $d_1$ on the finite energy space $\mathcal{E}^1(X,θ)$, making it a complete geodesic metric space. This construction is potentially more rigid compared to its analog f…

Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Ampère type arising in Kähler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of …

We develop a method for preserving pseudoholomorphic curves in contact 3-manifolds under surgery along transverse links. This makes use of a geometrically natural boundary value problem for holomorphic curves in a 3-manifold with stable Hamiltonian structure, where the boundary conditions are defined by 1-parameter fam…

Study complex Monge-Ampère equations on compact Kähler manifolds.

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $L^q$ regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of $\mathbb{R}^n$, which imply asymptotic behavior of the solutions at i…

Study collapsing geometry of hyperkähler 4-manifolds and prove conjectures.

The paper studies strong topologies for complex Monge-Ampère equations on Kähler manifolds.

Critical metrics on four-dimensional manifolds are either Einstein or product of two-dimensional manifolds.

Let $(X,ω)$ be a compact normal Kähler space, with Hodge metric $ω$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical Kähler metrics, we introduce a geodesic metric structure on $\mathcal H_ω(X)$, the space of Kähler potentials, whose completion is the fin…

The moduli space of static finite energy solutions to Ward's integrable chiral model is the space $M_N$ of based rational maps from $\CP^1$ to itself with degree $N$. The Lagrangian of Ward's model gives rise to a Kähler metric and a magnetic vector potential on this space. However, the magnetic field strength vanishes…

The paper studies quaternionic Monge-Ampère equations in weighted energy classes.

Establishes convexity and coercivity of K-energy functional for complex tori.

We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out tha…

The paper proves existence of solutions to the Allen-Cahn equation on certain Riemannian manifolds.

The paper derives Liouville theorems for various generalized maps on Riemannian manifolds.

We study the H^n-Yamabe constants of Riemannian products (H^n \times M^m, g_h^n +g), where (M,g) is a compact Riemannian manifold of constant scalar curvature and g_h^n is the hyperbolic metric on H^n. Numerical calculations can be carried out due to the uniqueness of (positive, finite energy) solutions of the equation…

Introduces non-Archimedean metrics for pseudoeffective classes on Kähler manifolds.

Geodesic rays prove key aspects of cscK metrics existence and stability.

In this paper we propose a geometrization of the non-relativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamic-dependent metric tensor on the principal manifold, such that the…

Study generalizes Möbius energy to non-smooth sets in arbitrary dimensions.

The paper solves a Nirenberg problem on half spheres, finding multiple blow-ups.

We construct finite energy instanton connection on $R^4$ which are periodic in two directions via an analogue of the Nahm transform for certain singular solutions of Hitchin's equations defined over a 2-torus.

We compute eta invariants of various Dirac type operators on circle bundles over Riemann surfaces via two approaches: an adiabatic approach based on the results of Bismut-Cheeger-Dai and a direct elementary one. These results, coupled with some delicate spectral flow computations are then used to determine the virtual …

Researchers create BPS monopoles with any desired symmetry breaking.

We investigate probability measures with finite pluricomplex energy. We give criteria insuring that a given measure has finite energy and test these on various examples. We show that this notion is a biholomorphic but not a bimeromorphic invariant.

In this paper, we study harmonic functions on weighted manifolds and harmonic maps from weighted manifolds into Hadamard spaces introduced by Korevaar and Schoen. We prove Liouville theorems for these harmonic maps with finite energy.