arXiv research

Sharp inequalities for $J$ functional on Kahler metrics.

Researchers found all special metrics in 4D for certain curvature functionals.

We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obt…

Paper characterizes embeddability of function spaces into $L_p$-type RKBS via metric entropy.

Paper finds critical metrics with pinched curvature are geodesic balls.

In this paper, we investigate critical points of the Laplacian's eigenvalues considered as functionals on the space of Riemmannian metrics or a conformal class of metrics on a compact manifold. We obtain necessary and sufficient conditions for a metric to be a critical point of such a functional. We derive specific con…

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

Study critical metrics on manifolds with boundary using integral and boundary estimates.

The study proves non-existence of concave functions on specific metric spaces.

Geodesics found in a metric space of m-subharmonic functions.

Study critical metrics on manifolds, proving specific isometries.

Study continuity of complex Sobolev functions, with applications to Kaehler metrics.

New functionals defined for free boundary minimal submanifolds in higher dimensions.

A $W^{1,p}$-metric on an $n$-dimensional closed Riemannian manifold naturally induces a distance function, provided $p$ is sufficiently close to $n$. If a sequence of metrics $g_k$ converges in $W^{1,p}$ to a limit metric $g$, then the corresponding distance functions $d_{g_k}$ subconverge to a limit distance function …

New estimates for Green's functions in varying Kähler metrics.

The paper proves the existence of singular cscK metrics on smoothable varieties.

Defines metrics and Einstein tensors on Riemannian manifolds, proving vanishing for non-commutative two-torus.

In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the $γ_k$ functions on the space of its hermitian metrics.

The paper establishes Sobolev inequalities between Riemannian metrics and their distance functions.

In this paper we prove rigidity results on critical metrics for quadratic curvature functionals, involving the Ricci and the scalar curvature, on the space of Riemannian metrics with unit volume. It is well-known that Einstein metrics are always critical points. The purpose of this article is to show that, under some c…

Study examines metrics and functionals for Hodge-Dirac operator on manifolds.

Constructs metrics with negative curvature on specific manifold types.

Proves existence of weighted-cscK metrics on Kähler manifolds.

New framework for studying eigenvalue functionals of metrics.

Study proposes a functional for LCK metrics on complex manifolds.

Study Hilbert's projective metric for bounded growth functions leading to Sinkhorn's algorithm convergence.

Classifies Kähler metrics with constant holomorphic curvature.

In this article, we investigate the geometry of critical metrics of the volume functional on an $n$-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifol…

New approach finds Kähler metrics on compact complex manifolds.

Constructs conformal metrics with negative curvature on manifolds with boundary.

Compact embeddings for invariant functions in metric-measure spaces.

Study on stability of ALE Ricci-flat metrics using a modified Perelman's λ-functional.

We study the Euler-Lagrange equation for several natural functionals defined on a conformal class of almost Hermitian metrics, whose expression involves the Lee form $θ$ of the metric. We show that the Gauduchon metrics are the unique extremal metrics of the functional corresponding to the norm of the codifferential of…

Given a simply connected compact generalized flag manifold M together with its invariant Kähler Einstein metric g, we investigate the functional given by the first eigenvalue of the Hodge Laplacian on smooth functions restricted to the space of invariant Kähler metrics. We give sufficient and necessary conditions so th…

Solves Yamabe problem for 3D metrics of Sobolev class $W^{2,q}$.

We introduce the coupled Ricci-Calabi functional and the coupled H-functional which measure how far from a coupled Kähler-Einstein metric in the sense of Hultgren-Witt Nyström. We first give corresponding moment weight type inequalities which estimate each functional in terms of algebraic invariants. Secondly, we give …

Let $M$ be a compact complex manifold admitting a Kähler structure. A conformally Kähler, Einstein-Maxwell metric (cKEM metric for short) is a Hermitian metric $\tilde{g}$ on $M$ with constant scalar curvature such that there is a positive smooth function $f$ with $g = f^2 \tilde{g}$ being a Kähler metric and $f$ being…

Paper calculates third coefficient in Kaehler-Einstein metric expansion.

Proves limit curve theorem for incomplete metric spaces, applies to null distance in Lorentzian manifolds.

For singular metrics, there is no Quillen metric formalism on cohomology determinant. In this paper, we develop an admissible theory, with which the arithmetic Deligne-Riemann-Roch isometry can be established for singular metrics. As an application, we first study Weil-Petersson metrics and Takhtajan-Zograf metrics on …

Metric functions for phoneme perception capture the similarity structure among phonemes in a given language and therefore play a central role in phonology and psycho-linguistics. Various phenomena depend on phoneme similarity, such as spoken word recognition or serial recall from verbal working memory. This study prese…

Paper introduces a new cosmological volume function and its properties.

It is a well-known fact that on a bounded spectral interval the Dirac spectrum can be described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article we extend this result to a global version. We think of the spectrum of a Dirac operator as a function from the int…

We show for a certain class of operators $A$ and holomorphic functions $f$ that the functional calculus $A\mapsto f(A)$ is holomorphic. Using this result we are able to prove that fractional Laplacians $(1+Δ^g)^p$ depend real analytically on the metric $g$ in suitable Sobolev topologies. As an application we obtain loc…

We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenböck type formula for critical metrics of the volume functional on four-dimensional manifolds. As an application, we obtain a clas…

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, consta…

We show a general relation between the spatially disjoint product of probability density functions and the sum of their Fisher information metric tensors. We then utilise this result to give a method for constructing the probability density functions for an arbitrary Riemannian Fisher information metric tensor. We note…

In this paper, we study Mabuchi metrics on Fano manifolds. We prove that Mabuchi metrics exist if the modified Ding functional is proper modulo a reductive subgroup of its automorphism group. On the other hand, the inverse that Mabuchi metrics implies the properness is obtained by using Darvas-Rubinstein's properness p…