Extremal almost-Kähler metrics

Extremal Kaehler-Einstein metric for two-dimensional convex bodies

Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{ä}hler-Einstein equation $e^{-Φ} = \det D^2 Φ$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2 Φ$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2 Φ$ for arbitrary $K$ is uniformly bounded by $\frac{n-1}{4(n+1)}$ and verify this conjecture in the two-dimensional case. The general case remains open.

Stability under deformations of extremal almost-Kähler metrics in dimension 4

Given a path of almost-Kähler metrics compatible with a fixed symplectic form on a compact 4-manifold such that at time zero the almost-Kähler metric is an extremal Kähler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-Kähler metrics compatible with the same symplectic form, such that at each time the induced almost-complex structure is diffeomorphic to the one induced by the path.

A note on extremal toric almost Kähler metrics

An almost Kähler structure is {\it extremal} if the Hermitian scalar curvature is a Killing potential [29]. When the almost complex structure is integrable it coincides with extremal Kähler metric in the sense of Calabi [8]. We observe that the existence of an extremal {\it toric} almost Kähler structure of involutive type implies uniform K-stability and we point out the existence of a formal solution of the Abreu equation for any angle along the invariant divisor. Applying the recent result of Chen--Cheng [10] and He [27], we conclude that the existence of a compatible extremal toric almost K…

Spectrum of hypersurfaces with small extrinsic radius or large $λ_1$ in euclidean spaces

In this paper, we prove that Euclidean hypersurfaces with almost extremal extrinsic radius or $λ_1$ have a spectrum that asymptotically contains the spectrum of the extremal sphere in the Reilly or Hasanis-Koutroufiotis Inequalities. We also consider almost extremal hypersurfaces which satisfy a supplementary bound on $v_M\|\B\|_α^n$ and show that their spectral and topological properties depends on the position of $α$ with respect to the critical value $\dim M$. The study of the metric shape of these extremal hypersurfaces will be done in \cite{AG1}, using estimates of the present paper.

关于拟统计流形上的 Kähler 和 anti-Kähler 结构的注记

Notes concerning Kähler and anti-Kähler structures on quasi-statistical manifolds

本文研究了拟统计流形上的近 Hermite 和近反 Hermite 结构，给出了近复结构可积性条件，并证明了其在特定条件下成为 Kähler、anti-Kähler 及拟 Kähler-Norden 流形的充要或必要条件。

Hypersurfaces with small extrinsic radius or large $λ_1$ in Euclidean spaces

We prove that hypersurfaces of $\R^{n+1}$ which are almost extremal for the Reilly inequality on $λ_1$ and have $L^p$-bounded mean curvature ($p>n$) are Hausdorff close to a sphere, have almost constant mean curvature and have a spectrum which asymptotically contains the spectrum of the sphere. We prove the same result for the Hasanis-Koutroufiotis inequality on extrinsic radius. We also prove that when a supplementary $L^q$ bound on the second fundamental is assumed, the almost extremal manifolds are Lipschitz close to a sphere when $q>n$, but not necessarily diffeomorphic to a sphere when $q…

Variational problems in conformal geometry

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 the Lee form. We prove that on compact complex surfaces, in every conformal class there exists a unique metric, up to multiplication by a constant, which is extremal for the functional given by the $L^2$-norm of $dJθ$, where $J$ denotes the complex structure. These extremal metr…

射影丛上的新极端凯勒度量

New extremal Kähler metrics on projective bundles

该研究证明射影化空间 $\boldsymbol{P}(E)$ 存在对合型兼容极端近凯勒度量等价于存在Calabi极端凯勒度量，利用对应关系简化方程，并应用于秩4向量丛的射影丛，证明其为Calabi梦想流形。

Local properties of almost-Riemannian structures in dimension 3

A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some set $\Zz$ called the singular set. Under the Hormander condition, a 3D almost-Riemannian structure still has a metric space structure, whose topology is compatible with the original topology of the manifold. Almost-Riemannian manifolds were deeply studied in dimension 2. In this paper we start the study of the 3D case which appear to be reacher with respect to the 2D case, due to the presence of abnormal extremals…

Canonical almost-Kähler metrics dual to general plane-fronted wave Lorentzian metrics

In the compact setting, Aazami and Ream \cite{Aazami:2022th} proved that Riemannian metrics dual to a class of Lorentzian metrics, called (compact) general plane-fronted waves, are almost-Kähler. In this note, we explain how to construct extremal and second-Chern-Einstein non-Kähler almost-Kähler metrics dual to those general plane-fronted waves.

The geometry of weakly selfdual Kahler surfaces

We study Kahler surfaces with harmonic anti-selfdual Weyl tensor. We provide an explicit local description, which we use to obtain the complete classification in the compact case. We give new examples of extremal Kahler metrics, including Kahler-Einstein metrics and conformally Einstein Kahler metrics. We also extend some of our results to almost Kahler 4-manifolds, providing new examples of Ricci-flat almost Kahler metrics which are not Kahler.

Eigenvalue pinching on convex domains in space forms

In this paper, we show that the convex domains of the hyperbolic space which are almost extremal for the Faber-Krahn or the Payne-Polya-Weinberger inequalities are close to geodesic balls. Our proof is also valid in other space forms and allows us to recover known results in Euclidean space and on the sphere.

Sharp quantitative isoperimetric inequalities in the $L^1$ Minkowski plane

We prove that a plane domain which is almost isoperimetric (with respect to the $L^1$ metric) is close to a square whose sides are parallel to the coordinates axis. Closeness is measured either by $L^\infty$ Haussdorf distance or Fraenkel asymmetry. In the first case, we determine the extremal domains.

无穷维中动量映射的范数平方及其在Kähler几何与辛联络中的应用

Norm-squared of the momentum map in infinite dimensions with applications to Kähler geometry and symplectic connections

本研究率先在无穷维中以动量映射范数平方为工具，推导其临界点Hessian性质与稳定子分解，应用于辛微分同胚群作用的两类空间，推广Calabi结果并发现非等变带来的新现象，还通过预量子化得到辛微分同胚群的新中心扩张。