Optimal geometric estimates for compact K\"ahler manifolds of a Nash entropy bound
We prove Sobolev-type inequality and local volume noncollapsing with optimal exponents for compact K\"ahler manifolds of uniformly bounded $q$-Nash entropy.
math.DG↗
arXiv subjects
Yashan Zhang
Publications and source records attributed to Yashan Zhang.
Green's function estimates for compact Kähler manifolds and applications
Recent works of Guo-Phong-Song-Sturm established for compact Kähler manifolds (even for Kähler spaces of specific singularities) a variety of geometric estimates depending on an upper bound of $L^{1+ε}$ or $L^1(\log L)^{n+ε}$ norms of the volume density but not on any curvature bound, in which a key ingredient is a uniform integral estimate for Green's function. Motivated by their results and further applications, in this paper we shall prove an improved (nearly optimal) integral estimate for Green's function under $L^{1+ε}$ volume density condition, and then apply it to obtain improved global geometric estimates. For instance, one of our results states that the $k$th eigenvalue of Laplacian operator $λ_k\ge c\cdot k^{\frac{1}{n}}(\log k)^{-3}$, where $n$ is the complex dimension of the Kähler manifold and $c$ depends on $n$ and $L^{1+ε}$ norm of the volume density. Also, our results can be applied to the long-time or volume-noncollapsing finite-time Kähler-Ricci flow on compact Kähler manifolds and to a general Kähler family to further extend previous works of Guo-Phong-Song-Sturm, Guedj-Tô and Vu.
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A note on Teissier problem for nef classes
Teissier problem aims to characterize the equality case of Khovanskii-Teissier type inequality for $(1,1)$-classes on a compact Kähler manifold. When each of the involved $(1,1)$-classes is assumed to be nef and big, this problem has been solved by the previous works of Boucksom-Favre-Jonsson, Fu-Xiao and Li. In this note, we shall settle the case that the involved $(1,1)$-classes are just assumed to be nef. We also extend the results to some settings where some of the $(1,1)$-classes are not necessarily nef. By constructing examples, it is shown that our results are optimal.
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Positivity in Foliated Manifolds and Geometric Applications
We introduce the notion of positivity for a real basic $(1,1)$ class in basic Bott-Chern cohomology group on foliated manifolds, and study the relationship between this positivity and the negativity of transverse holomorphic sectional curvature and give some geometric applications.
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On almost quasi-negative holomorphic sectional curvature
A recent celebrated theorem of Diverio-Trapani and Wu-Yau states that a compact Kähler manifold admitting a Kähler metric of quasi-negative holomorphic sectional curvature has an ample canonical line bundle, confirming a conjecture of Yau. In this paper we shall consider a natural notion of almost quasi-negative holomorphic sectional curvature and extend this theorem to compact Kähler manifolds of almost quasi-negative holomorphic sectional curvature. We also obtain a gap-type theorem for the inequality $\int_Xc_1(K_X)^n>0$ in terms of the holomorphic sectional curvature. In the discussions, we introduce a capacity notion for the negative part of holomorphic sectional curvature, which plays a key role in studying the relation between the almost quasi-negative holomorphic sectional curvature and ampleness of the canonical line bundle.
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On higher-rank Khovanskii-Teissier inequalities
We shall discuss a higher-rank Khovanskii-Teissier inequality, generalizing a theorem of Li. In the course of the proof, we develop new Hodge-Riemann bilinear relations in certain mixed settings, which in themselves slightly extend the existing results and imply new Khovanskii-Teissier type inequalities and log-concavity results.
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Integral inequalities for holomorphic maps and applications
We derive some integral inequalities for holomorphic maps between complex manifolds. As applications, some rigidity and degeneracy theorems for holomorphic maps without assuming any pointwise curvature signs for both the domain and target manifolds are proved, in which key roles are played by total integration of the function of the first eigenvalue of second Ricci curvature and an almost nonpositivity notion for holomorphic sectional curvature introduced in our previous work. We also apply these integral inequalities to discuss the infinite-time singularity type of the Kähler-Ricci flow. The equality case is characterized for some special settings.
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Holomorphic sectional curvature, nefness and Miyaoka-Yau type inequality
On a compact Kähler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kähler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kähler manifold of almost nonpositive holomorphic sectional curvature has a nef canonical line bundle, contains no rational curves and satisfies some Miyaoka-Yau type inequalities. In the course of the discussions, we attach a real value to any fixed Kähler class which, up to a constant factor depending only on the dimension of manifold, turns out to be an upper bound for the nef threshold.
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Local curvature estimates of long-time solutions to the Kähler-Ricci flow
We study the local curvature estimates of long-time solutions to the normalized Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundles. Using these estimates, we prove that on such a manifold, the set of singular fibers of the semi-ample fibration on which the Riemann curvature blows up at time-infinity is independent of the choice of the initial Kähler metric. Moreover, when the regular fiber of the semi-ample fibration is not a finite quotient of a torus, we determine the exact curvature blow-up rate of the Kähler-Ricci flow near the regular fiber.
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Infinite-time singularity type of the Kähler-Ricci flow II
On a compact Kähler manifold with semi-ample canonical line bundle and Kodaira dimension one, we observe a relation between the infinite-time singularity type of the Kähler-Ricci flow and the characteristic indexes of singular fibers of the semi-ample fibration.
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The continuity method on Fano fibrations
We study finite-time collapsing limits of the continuity method. When the continuity method starting from a rational initial Kähler metric on a projective manifold encounters a finite-time volume collapsing, this projective manifold admits a Fano fibration over a lower dimensional base. In this case, we prove the continuity method converges to a singular Kähler metric on the base in the weak sense; moreover, if the base is smooth and the fibration has no singular fibers, then the convergence takes place in Gromov-Hausdorff topology.
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On a twisted conical Kähler-Ricci flow
In this paper, we discuss diameter bound and Gromov-Hausdorff convergence of a twisted conical Kähler-Ricci flow on the total spaces of some holomorphic submersions. We also observe that, starting from a model conical Kähler metric with possibly unbounded scalar curvature, the conical Kähler-Ricci flow will instantly have bounded scalar curvature for $t>0$, and the bound is of the form $\frac{C}{t}$. Several key results will be obtained by direct arguments on the conical equation without passing to a smooth approximation. In the last section, we present several remarks on a twisted Kähler-Ricci flow and its convergence.
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Collapsing limits of the Kähler-Ricci flow and the continuity method
We consider the Kähler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the Kähler-Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Gromov-Hausdorff topology to the metric completion of the regular part of generalized Kähler-Einstein current on the base, which is a compact length metric space homeomorphic to the base. The analogue results for the continuity method on such Calabi-Yau fibrations are also obtained. Moreover, we show the continuity method starting from a suitable Kähler metric on the total space of a Fano fibration with one dimensional base converges in Gromov-Hausdorff topology to a compact metric on the base. During the proof, we show the metric completion of the regular part of a generalized Kähler-Einstein current on a Riemann surface is compact.
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Infinite-time singularity type of the Kähler-Ricci flow
For the Kähler-Ricci flow on a compact Kähler manifold with semi-ample canonical line bundle, we prove the singularity type at infinity does not depend on the choice of the initial metric. We also provide new simple proofs for some existing classification results on infinite-time singularity type of the Kähler-Ricci flow.
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The continuity method on minimal elliptic Kähler surfaces
We prove that, on a minimal elliptic Kähler surface of Kodaira dimension one, the continuity method introduced by La Nave and Tian in \cite{LT} starting from any initial Kähler metric converges in Gromov-Hausdorff topology to the metric completion of the generalized Kähler-Einstein metric on its canonical model constructed by Song and Tian in \cite{ST06}.
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A note on conical Kähler-Ricci flow on minimal elliptic Kähler surfaces
We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical Kähler-Ricci flow on a minimal elliptic Kähler surface converges in the sense of currents to a generalized conical Kähler-Einstein on its canonical model. Moreover, the convergence takes place smoothly outside the singular fibers and the chosen divisor.
math.DG↗
On the change of variables formula for multiple integrals
In this paper, we develop an elementary proof of the change of variables in multiple integrals. Our proof is based on an induction argument. Assuming the formula for (m-1)-integrals, we define the integral over hypersurface in Rm, establish the divergent theorem and then use the divergent theorem to prove the formula for m-integrals. In addition to its simplicity, an advantage of our approach is that it yields the Brouwer Fixed Point Theorem as a corollary.
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