This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills(HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with K?hler metri...This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills(HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with K?hler metrics ωε when ε → 0. Here, ωε are flat and have areas ε and ε-1 on the two elliptic curves, respectively.A family of Hermitian metrics on the vector bundle are explicitly constructed and with respect to them, the HYM metrics are normalized. We then compare the family of normalized HYM metrics with the family of constructed Hermitian metrics by doing estimates. We get the higher order estimates as long as the C^0-estimate is provided. We also get the estimate of the lower bound of the C^0-norm. If the desired estimate of the upper bound of the C^0-norm can be obtained, then it would be shown that these two families of metrics are close to arbitrary order in ε in any Cknorms.展开更多
On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of...On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of the Riemannian scalar curvature, norms of the components of the covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we use them to get characterization results of the K?hler metric, the balanced metric, the locally conformal K?hler metric or the k-Gauduchon metric. As corollaries, we show partial results related to a problem given by Lejmi and Upmeier(2020) and a conjecture by Angella et al.(2018).展开更多
We give the extension formulae on almost complex manifolds and give decompositions of the extension formulae.As applications,we study(n,0)-forms,the(n,0)-Dolbeault cohomology group and(n,q)-forms on almost complex man...We give the extension formulae on almost complex manifolds and give decompositions of the extension formulae.As applications,we study(n,0)-forms,the(n,0)-Dolbeault cohomology group and(n,q)-forms on almost complex manifolds.展开更多
A Hermitian curvature flow on a compact Calabi-Yau manifold is proposed and a regularity result is obtained.The solution of the flow,if exists,is a balanced HermitianEinstein metric.
基金supported by National Natural Science Foundation of China (Grant Nos. 11871016, 11421061 and 11025103)
文摘This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills(HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with K?hler metrics ωε when ε → 0. Here, ωε are flat and have areas ε and ε-1 on the two elliptic curves, respectively.A family of Hermitian metrics on the vector bundle are explicitly constructed and with respect to them, the HYM metrics are normalized. We then compare the family of normalized HYM metrics with the family of constructed Hermitian metrics by doing estimates. We get the higher order estimates as long as the C^0-estimate is provided. We also get the estimate of the lower bound of the C^0-norm. If the desired estimate of the upper bound of the C^0-norm can be obtained, then it would be shown that these two families of metrics are close to arbitrary order in ε in any Cknorms.
基金supported by National Natural Science Foundation of China(Grant Nos.10831008,11025103 and 11501505)。
文摘On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of the Riemannian scalar curvature, norms of the components of the covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we use them to get characterization results of the K?hler metric, the balanced metric, the locally conformal K?hler metric or the k-Gauduchon metric. As corollaries, we show partial results related to a problem given by Lejmi and Upmeier(2020) and a conjecture by Angella et al.(2018).
基金supported by National Natural Science Foundation of China(Grant No.11871016)。
文摘We give the extension formulae on almost complex manifolds and give decompositions of the extension formulae.As applications,we study(n,0)-forms,the(n,0)-Dolbeault cohomology group and(n,q)-forms on almost complex manifolds.
基金supported by the National Natural Science Foundation of China(No.11871016)。
文摘A Hermitian curvature flow on a compact Calabi-Yau manifold is proposed and a regularity result is obtained.The solution of the flow,if exists,is a balanced HermitianEinstein metric.
