Gábor Székelyhidi, Valentino Tosatti, Ben Weinkove
We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.
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The prescribed Gauduchon scalar curvature problem in almost Hermitian geometry
该研究针对近厄米流形的指定Gauduchon标量曲率问题，通过共形变化推导将其转化为带指数非线性的半线性偏微分方程，用上下解方法证明解的存在性与Gauduchon度相关常数的符号有关，负号时给出充要条件，还复现了三类相关Yamabe问题。
$L^\infty$ estimate for the potential of quaternionic Gauduchon metric with prescribed volume form
本文受四元数 Calabi 猜想启发，在紧致 $SL(n,\mathbb{H})$ 流形上引入四元数 Gauduchon 猜想，研究具有指定体积形式的四元数 Gauduchon 度量的存在性，将其转化为二阶完全非线性椭圆方程并建立势函数的 $L^\infty$ 估计。
Prescribed Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree
本文研究具有负Gauduchon度的紧Hermitian流形上预定Chern数量曲率问题，利用几何流收敛性在非正及变号曲率函数情形下建立了存在性与不存在性结果。
Prescribed Chern scalar curvature flow on compact Hermitian manifolds with negative Gauduchon degree
针对负 Gauduchon 度紧 Hermitian 流形，提出统一流方法，并在共形类含平衡度量时给出保证流收敛至 Chern 标量曲率为 $f$ 的充分条件。
The paper is an attempt to resolve the prescribed Chern scalar curvature problem. We look for solutions within the conformal class of a fixed Hermitian metric. We divide the problem in three cases, according to the sign of the Gauduchon degree, that we analyse separately. In the case where the Gauduchon degree is negative, we prove that every non-identically zero and non-positive function is the Chern scalar curvature of a unique metric conformal to the fixed one. Moreover, if there exists a balanced metric with zero Chern scalar curvature, we prove that every smooth function changing sign wit…
Singular Gauduchon Conjecture
本文研究了Gauduchon猜想的奇点版本，通过适应Guedj-Lu的技术，得到了$C^{0}$估计，并推广了TW17的结果。
A note on the Chern-Yamabe flow on compact Hermitian manifolds with negative Gauduchon degree
本注记证明，在具有负Gauduchon次数的紧Hermitian流形上，当候选曲率函数非正且不恒为零时，Chern-Yamabe流收敛到规定Chern标量曲率问题的解。
Deformed Gauduchon-Yamabe problems on compact Hermitian manifolds
本文研究紧致 Hermitian 流形上 Gauduchon 联络的标量曲率形变，给出常曲率度量存在判据，并分析 Yamabe 不变量的全局依赖性及临界阈值应用。
Isosystolic Inequalities for Holomorphic Chains in $\mathbb{C}P^{n}$
该研究定义了复射影空间上埃尔米特度量的全纯k-收缩，证明Fubini-Study度量是Gauduchon度量中体积归一化全纯$(n-1)$-收缩的局部极小值，并构造出接近Fubini-Study度量、同调2-收缩由非全纯链实现的Gauduchon度量。
We establish existence of compact minimizers of the prescribed mean curvature problem with volume constraint in periodic media. As a consequence, we construct compact approximate solutions to the prescribed mean curvature equation. We also show convergence after rescaling of the volume-constrained minimizers towards a suitable Wulff Shape, when the volume tends to infinity.
Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity
本文证明了具 prescribed singularity 的复 Monge-Ampère 方程解的存在性与唯一性，去除了小无界轨迹假设，并证实了闭正 $(1,1)$-currents 体积对数凹性猜想，揭示了其与凸几何 Brunn-Minkowski 不等式的对应关系。
We solve the modified Kazdan-Warner problem of finding metrics with prescribed scalar curvature and unit total volume.
We study the solutions $u\in C^\infty(R^{2m})$ of the problem $(-Δ)^m u= Qe^{2mu}$, where $Q=\pm (2m-1)!$, and $V :=\int_{R^{2m}}e^{2mu}dx <\infty$, particularly when $m>1$. This corresponds to finding conformal metrics $g_u:=e^{2u}|dx|^2$ on $R^{2m}$ with constant Q-curvature $Q$ and finite volume $V$. Extending previous works of Chang-Chen, and Wei-Ye, we show that both the value $V$ and the asymptotic behavior of $u(x)$ as $|x|\to \infty$ can be simultaneously prescribed, under certain restrictions. When $Q=(2m-1)!$ we need to assume $V<vol(S^{2m})$, but surprisingly for $Q=-(2m-1)!$ the vo…
On Ricci forms of canonical metrics over noncompact complex manifolds
研究非紧复流形上里奇曲率相关的几何问题，包括负里奇曲率下紧致凯勒流形上凯勒-爱因斯坦度量的存在性，以及完备赫尔米特流形上给定里奇曲率的规范赫尔米特度量的存在性，提出构造阿费-爱因斯坦度量的方法。
In this paper,we obtain two results on closed Reimainnian manifold $M\times [0,T]$.When $T$ is small enough,to any prescribed scalar curvature, the existence and uniqueness of metrics are obtained on the volume element preserving deformation.When $T$ is large and the given scalar curvature is small enough,the same result holds.
We prove that a critical metric of the volume functional on a $4$-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form $\mathbb{R}^{4}$, $\mathbb{H}^{4}$ or $\mathbb{S}^{4}.$ Moreover, we provide an integral curvature estimate involving the Yamabe constant for critical metrics of the volume functional, which allows us to get a rigidity result for such critical metrics.
Let $\{α\}$ and $\{β\}$ be nef cohomology classes of bidegree $(1,\,1)$ on a compact $n$-dimensional Kähler manifold $X$ such that the difference of intersection numbers $\{α\}^n - n\,\{α\}^{n-1}.\,\{β\}$ is positive. We solve in a number of special but rather inclusive cases the quantitative part of Demailly's Transcendental Morse Inequalities Conjecture for this context predicting the lower bound $\{α\}^n-n\,\{α\}^{n-1}.\,\{β\}$ for the volume of the difference class $\{α-β\}$. We completely solved the qualitative part in an earlier work. We also give general lower bounds for the volume of $…
A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kaehler covering W, with the deck transform acting on W by homotheties. If compact, it admits a canonical Vaisman metric, due to Gauduchon. We show that a Hermitian Einstein-Weyl structure on a compact complex manifold is determined by its volume form. This result is a conformal analogue of Calabi's theorem stating the uniqueness of Kaehler metrics with a given volume form in a given Kaehler class. We prove that a solution of a conformal version of complex Monge-Ampere equation is unique. We conjecture that a Hermi…
We develop a theory of stable bundles and affine Hermitian-Einstein metrics for flat vector bundles over a special affine manifold (a manifold admitting an atlas whose gluing maps are all locally constant volume-preserving affine maps). Our paper presents a parallel to Donaldson-Uhlenbeck-Yau's proof of the existence of Hermitian-Einstein metrics on Kähler manifolds, and the extension of this theorem by Li-Yau to the non-Kähler complex case of Gauduchon metrics. Our definition of stability involves only flat vector subbundles (and not singular subsheaves), and so is simpler than the complex ca…
Riemannian metrics with prescribed volume and finite parts of Dirichlet spectrum
本文证明了在带光滑边界的紧流形上，可以构造黎曼度量同时实现规定的体积和任意给定的有限Dirichlet特征值序列。
Min-max construction of prescribed mean curvature hypersurfaces in noncompact manifolds
该研究建立了非紧流形中预定平均曲率超曲面的极小-极大理论，适用于紧集外不变号的预定函数，并借此证明了两类超曲面的新存在性结果。
Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal $G$-bundle $E_G$ over $M$ admits a Hermitian-Einstein structure if and only if $E_G$ is polystable. A polystable flat principal $G$--bundle over $M$ admits a unique Hermitian-Einstein connection. We also prove the existence and uniqueness of a Harder-Narasimhan filtration for …
In this paper we consider surfaces of class $C^1$ with continuous prescribed mean curvature in a three-dimensional contact sub-Riemannian manifold and prove that their characteristic curves are of class $C^2$. This regularity result also holds for critical points of the sub-Riemannian perimeter under a volume constraint. All results are valid in the first Heisenberg group $\mathbb{H}^1$.
We consider smooth bounded surfaces with a smooth boundary and a prescribed background metric g_0. We now consider all metrics g conformal to g_0 which have a prescribed volume M. We now minimize the first eigenvalue of the Laplace operator of g over the metrics conformal to g_0 and having the prescribed volume. We show that this problem is equivalent to the study of the Composite Membrane Problem, a free boundary problem studied earlier by the author and his collaborators in all dimensions. Thus complete answers, existence of the limit metric, regularity of the minimizing eigenfunction and va…
Given a smooth compact hypersurface $M$ with boundary $Σ=\partial M$, we prove the existence of a sequence $M_j$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $σ_k(M_j)$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_j$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$, while the principal curvatures of the boundary remain unchanged.
We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the…
We prove the existence of plurisubharmonic functions with prescribed logarithmic singularities on complex 3-folds equipped with a nef class of positive volume. We prove the same result for rational classes on Moishezon n-folds.
We study the localization of sets with constant nonlocal mean curvature and prescribed small volume in a bounded open set with smooth boundary, proving that they are {\em sufficiently close} to critical points of a suitable non-local potential. We then consider the fractional perimeter in half-spaces. We prove the existence of a minimizer under fixed volume constraint, showing some of its properties such as smoothness and symmetry, being a graph in the $x_N$-direction, and characterizing its intersection with the hyperplane $\{x_N=0\}$.
We prove that, among metrics on a compact quotient of $\left(\mathbb{H}^2\right)^n$ (product of hyperbolic planes) of prescribed total volume, the product of hyperbolic metrics has minimal volume entropy.
We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each volume spectrum is realized by the min-max value of certain relative homotopy class of sweepouts of boundaries of Caccioppoli sets. The main result follows by approximating such min-max value using the min-max theory for hypersurfaces with prescribed mean curvature established b…
Volumes of moduli spaces of flat surfaces
本文研究了具有预定锥奇点的平坦曲面模空间，证明了其体积有限，并通过对其穿孔曲面的 Euler 示性数进行归纳，证明了体积在几乎所有奇点阶数下均可显式计算。
Fully non-linear elliptic equations on noncompact complex manifolds
本文研究非紧复流形上非线性椭圆方程的先验估计与存在性，应用包括构造具有给定体积形式的凯勒度量及爱因斯坦度量。
We study metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$(-Δ)^\frac{n}{2}w=e^{nw}-cδ_{0} \text{ on } \mathbb R^n,$$ under a finite volume condition. We analyze the asymptotic behaviour at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.
We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally.
Boundary exceptional sets for radial limits of superharmonic functions on non-positively curved Harmonic manifolds of purely exponential volume growth
本文研究了纯指数体积增长非正曲率Harmonic流形上，函数沿径向测地射线发散的边界例外集大小，利用Poisson积分和Green位势获得了Hausdorff维数的尖锐界限与上界，并结合Riesz分解定理得到了正上调和函数例外集维数的主要结果。
Given a smooth compact manifold with boundary, we study variational properties of the volume functional and of the area functional of the boundary, restricted to the space of the Riemannian metrics with prescribed curvature. We obtain a sufficient and necessary condition for a metric to be a critical point. As a by-product, a very natural analogue of V-statics metrics is obtained. In the second part, using the Yamabe invariant in the boundary setting, we solve the Kazdan-Warner-Kobayashi problem in a compact manifold with boundary. For several cases, depending on the signal of the Yamabe invar…
In a compact orbifold, for small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.
We derive a formula of Chern-Gauss-Bonnet type for the Euler characteristic of a four dimensional manifold-with-boundary in terms of the geometry of the Loewner-Nirenberg singular Yamabe metric in a prescribed conformal class. The formula involves the renormalized volume and a boundary integral. It is shown that if the boundary is umbilic, then the sum of the renormalized volume and the boundary integral is a conformal invariant. Analogous results are proved for asymptotically hyperbolic metrics in dimension four for which the second elementary symmetric function of the eigenvalues of the Scho…
The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. In this paper, we localize a condition satisfied by such stationary points to smooth bounded domains. The condition involves a generalization of the static equations, and we interpret solutions (and their boundary values) of this equation variationally. On…
Let M and N be closed n-dimensional manifolds, and equip N with a volume form σ. Let μbe an exact n-form on M. Arnold then asked the question: When can one find a map f:;N such that f*σ=μ. In 1973 Eliashberg and Gromov showed that this problem is, in a deep sense, trivial: It satisfies an h-principle, and whenever one can find a bundle map f_bdl:T M to T N which is degree 0 on the base and induces μone can homotop this map to a solution f. That is if the naive topological conditions are satisfied on can find a solution. There is no further interesting geometry in the problem. We show the corre…
Given $d\in \mathbb{N}$, $g\in \mathbb{N} \cup\{0\}$, and an integral vector $κ=(k_1,\dots,k_n)$ such that $k_i>-d$ and $k_1+\dots+k_n=d(2g-2)$, let $Ω^d\mathcal{M}_{g,n}(κ)$ denote the moduli space of meromorphic $d$-differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $κ$. We show that $Ω^d\mathcal{M}_{g,n}(κ)$ carries a canonical volume form that is parallel with respect to its affine complex manifold structure, and that the total volume of $\mathbb{P}Ω^d\mathcal{M}_{g,n}(κ)=Ω^d\mathcal{M}_{g,n}/\mathbb{C}^*$ with respect to the measure induced by t…
We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a nondegenerate critical point of the scalar curvature of the Riemannian manifold was required.
Finite Volume Einstein Finsler Warped Product Manifolds of Non-positive or Non-negative Scalar Curvature
本文研究了非对称芬斯勒流形的有限体积爱因斯坦扭曲积结构，并探讨了其在相对论理论中的应用。
Two properties of optimisers for the reverse isoperimetric problem
研究黎曼流形中在给定半径球内自由滚动且固定体积的凸集，证明无C^2光滑的周长最大化者，并证明C^2光滑区域的最小主曲率为常数。
Generic Uniqueness of Isoperimetric Regions in Arbitrary Dimension
本文证明在任意维闭流形上，对一般黎曼度量，给定体积分数（除半体积外）的等周区域唯一；半体积时恰有两个极小化子，且证明无需边界正则性或非退化条件。
It is shown that any affine toric variety Y, which is Q-Gorenstein, admits a conical Ricci flat Kahler metric, which is smooth on the regular locus of Y. The corresponding Reeb vector is the unique minimizer of the volume functional on the Reeb cone of Y. The case when the vertex point of Y is an isolated singularity was previously shown by Futaki-Ono-Wang. The proof is based on an existence result for the inhomogeneous Monge-Ampere equation in real Euclidean space with exponential right hand side and prescribed target given by a proper convex convex, combined with transversal a priori estimat…
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