Gabriella Clemente
We study functionals on the space of almost complex structures on a compact $\mathbb{C}$-manifold, whose variational properties could be used to tackle Yau's Challenge.
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Godbillon-Vey type functional for almost contact manifolds
本文将 Godbillon-Vey 泛函引入三维近切触流形，推导出保持 Reeb 向量场的 Euler-Lagrange 方程，并构造了具有双重扭曲积结构且属于 Chinea-Gonzalez 分类 $C_{5}\oplus C_{12}$ 类的临界解。
Energy functionals on almost Kähler manifolds: I
本文研究了几乎凯勒流形上的能量泛函，推广了Aubin泛函之间的不等式。
We establish plurisubharmonicity of the envelope of Poisson and Lelong functionals on almost complex manifolds. That is, we generalize the corresponding results for complex manifolds and almost complex manifolds of complex dimension two. We also provide some applications to the regularization of J-plurisubharmonic functions and to the characterization of compact psh-hulls by pseudoholomorphic discs.
The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce the concept of Banach statistical limit functional and the notion of GAS convergence mainly depends on the existence of Banach statistical limit functional. We prove the existence of Banach statistical limit functional. Then we have shown the existence of a GAS convergent sequenc…
If a smooth compact 4-manifold M admits a Kaehler-Einstein metric g of positive scalar curvature, Gursky showed that its conformal class [g] is an absolute minimizer of the Weyl functional among all conformal classes with positive Yamabe constant. Here we prove that, with the same hypotheses, [g] also minimizes of the Weyl functional on a different open set of conformal classes, most of which have negative Yamabe constant. An analogous minimization result is then proved for Einstein metrics g which are Hermitian, but not Kaehler.
We study a variational problem on a smooth manifold with a decomposition of the tangent bundle into $k>2$ subbundles (distributions), namely, we consider the integrated sum of their mixed scalar curvatures as a functional of adapted pseudo-Riemannian metric (keeping the pairwise orthogonality of the distributions) and contorsion tensor, defining a linear connection. This functional allows us to generalize the class of Einstein metrics in the following sense: if all of the distributions are one-dimensional, then it coincides with the geometrical part of the Einstein-Hilbert action restricted to…
Hermitian Calabi functional in complexified orbits
本研究针对紧致辛流形上与辛形式相容的殆复结构空间，推导了极值殆凯勒度量处埃尔米特卡拉比泛函的黑塞显式公式，证明其在复化轨道临界点上半正定并得到相关推论，同时验证了埃尔米特卡拉比流的弱抛物性。
Let $(M,I)$ be an almost complex 6-manifold. The obstruction to integrability of almost complex structure (so-called Nijenhuis tensor) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold is called nearly Kaehler if it admits a Hermitian form $ω$ such that $\nabla(ω)$ is totally antisymmetric, $\nabla$ being the Levi-Civita connection. We show that a nearly Kaehler metric on a given almost complex 6-manifold with non-degenerate Nijenhuis tensor is unique (up to a constant). We interpret the nearly…
The infimum of the Weyl functional is shown to be surprisingly small on many compact 4-manifolds that admit positive-scalar-curvature metrics. Results are also proved that systematically compare the scalar and self-dual Weyl curvatures of certain almost-Kaehler 4-manifolds.
In this article we apply a Bochner type formula to show that on a compact conformally flat riemannian manifold (or half-conformally flat in dimension 4) certain types of orthogonal almost-complex structures, if they exist, give the absolute minimum for the energy functional. We give a few examples when such minimizers exist, and in particular, we prove that the standard almost-complex structure on the round S^6 gives the absolute minimum for the energy. We also discuss the uniqueness of this minimum and the extension of these results to other orthogonal G-structures.
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…
Regularity for free boundary surfaces minimizing degenerate area functionals
本文针对退化加权面积泛函的近似极小值点，建立了自由边界的ε-正则性定理，证明了边界为正则曲面且正交接触边界，并给出了奇异集的Hausdorff维数上界。
Łojasiewicz inequalities near simple bubble trees
针对闭合非球面曲面上的常数 H-泛函，本文证明临界点间隙现象及近临界点序列趋近首个非平凡泡泡树时的 Łojasiewicz 不等式，并给出 Hilbert 空间中相应的充分条件。
A semicontinuous semifinite trace is constructed on the C*-algebra generated by the finite propagation operators acting on the L^2-sections of a hermitian vector bundle on an amenable open manifold of bounded geometry. This trace is the semicontinuous regularization of a functional already considered by J. Roe. As an application, we show that, by means of this semicontinuous trace, Novikov-Shubin numbers for amenable manifolds can be defined (cf. math.OA/9802015 for an alternate definition).
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