复 Hessian 方程的极小斜率与起泡现象
Minimal slopes and bubbling for complex Hessian equations
AI总结：
本研究针对复 Hessian 方程在 Nakai 准则不成立的不稳定情形，提出寻找典范解的研究计划，证明 Kähler 曲面上 J-方程和形变 Hermitian Yang-Mills 方程存在唯一带奇性的典范解，并揭示 J-方程的起泡现象。
摘要：
一大类复 Hessian 方程光滑解的存在性，与 Kähler 流形上相交数的非线性 Nakai 型准则相关。这类 Nakai 准则可解释为一种斜率稳定性条件，类似于 Kähler 流形上 Hermitian 向量丛的斜率稳定性。在本研究中，我们启动了一项研究计划，旨在当 Nakai 准则不成立的不稳定情形下，寻找这类方程的典范解。据推测，这类解应作为自然抛物流的极限出现，且应为相应矩映射能量泛函的极小元。我们针对曲面上的 J-方程和形变 Hermitian Yang-Mills 方程，以及一些带对称性的例子，实施了我们的方法。我们证明，在 Kähler 曲面上的不稳定情形中，这两个方程始终存在唯一的典范解。这类带奇异性的典范解还被证明是某些射影丛上相应 J-流和余切流的极限。我们进一步通过构造矩映射能量泛函的极小化序列，展示了 J-方程的起泡现象，这些序列的 Gromov-Hausdorff 极限是奇异代数空间。
英文摘要：
The existence of smooth solutions to a broad class of complex Hessian equations is related to nonlinear Nakai type criteria on intersection numbers on Kahler manifolds. Such a Nakai criteria can be interpreted as a slope stability condition analogous to the slope stability for Hermitian vector bundles over Kahler manifolds. In the present work, we initiate a program to find canonical solutions to such equations in the unstable case when the Nakai criteria fails. Conjecturally such solutions should arise as limits of natural parabolic flows and should be minimisers of the corresponding moment-map energy functionals. We implement our approach for the J-equation and the deformed Hermitian Yang-Mills equation on surfaces and some examples with symmetry. We prove that there always exist unique canonical solutions to these two equations on Kahler surfaces in the unstable cases. Such canonical solutions with singularities are also shown to be the limits of the corresponding J-flow and the cotangent flow on certain projective bundles. We further present the bubbling phenomena for the J-equation by constructing minimizing sequences of the moment-map energy functionals, whose Gromov-Hausdorff limits are singular algebraic spaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
