Liouville and Calabi-Yau type theorems for complex Hessian equations
Liouville and Calabi-Yau type theorems for complex Hessian equations
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DOI：
10.1353/ajm.2017.0009
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发表时间：
2012-03
影响因子：
1.7
通讯作者：
S. Dinew;S. Kołodziej
中科院分区：
文献类型：
--
作者：
S. Dinew;S. Kołodziej
证明了具有有界梯度的整体极大$m$-次调和函数在有界梯度下的Liouvile型定理。这个结果，再加上一个标准的爆破引理，给出了紧K-Ahler流形上复Hessian方程的一个(非显式)先验梯度估计。这终止了由Hou，Ma和Wu发起的求解这类流形上的非退化Hessian方程的程序。我们还利用我们以前的工作，在某些$L^p$中得到了在右端退化情况下的连续弱解，并在$p$上得到了一个锐界
We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in ${\Bbb C}^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (nonexplicit) a priori gradient estimate for the complex Hessian equation on a compact K\"ahler manifold. This terminates the program, initiated by Hou, Ma, and Wu, of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak solutions in the degenerate case for the right-hand side in some $L^p$, with a sharp bound on $p$