Liouville and Calabi-Yau type theorems for complex Hessian equations

By S. Dinew, S. Kołodziej. 2012

151 citations recorded by the source corpus.

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 $p$

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6d9cb26599a90496dcfbbc3e2c502d6c0124ba8f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The Neumann Problem for Hessian Equations (2015, methodology): “etry, which started from Chang-Gursky-Yang [5] and the related development by ([16], [23], [17], [36],[10]). In Kahler geometry, the Hessian equation was studied by Hou-Ma-Wu [18] and Dinew-Kolodziej [8]. The Yamabe problem on manifolds with boundary was ﬁrst studied by Escobar [9], he shows that (a”
Fully non-linear parabolic equations on compact Hermitian manifolds (2017, methodology): “A general existence result on compact Hermitian manifolds was recently obtained by Dinew and Kolodziej [10], Sun [47], and Székelyhidi [50].”
Fu–Yau Hessian equations (2018, methodology): “ns out to hold because there is no cancellation in the expression e2u −4α′eu|∇u|2. The estimate for 2 |∇u| and |ˆσ2(ω′)| can then be obtained respectively by applying the techniques of DinewKolodziej [8], and Chou-X.J. Wang [6], Hou-Ma-Wu [16], Guan [14], and the authors [22]. The approach in the pr”
A priori estimates for complex Hessian equations (2011, methodology): “and consider the Hessian equation on a compact Kähler manifold X , as in [Hou 2009; Hou et al. 2010; Dinew and Kołodziej 2012]:”
On a class of fully nonlinear elliptic equations on Hermitian manifolds (2013, methodology): “hler. Indeed, deriving gradient estimates for fully nonlinear equations on complex manifolds turns out to be a rather challenging and mostly open question. Only very recently were Dinew and Kolodziej [9] able to prove the gradient estimate using scaling techniques and Liouville type theorems for the”
Gauduchon metrics with prescribed volume form (2015, unknown): “A blow-up argument as in [34, §6] combined with a Liouville theorem [34, §5] (see also [8], [39], [40]), shows that supM |∇u|(2)α6C, and so we get a uniform bound |∆u|6C.”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, methodology): “hese existing estimates depend on at least two derivatives of ψ. Theorem 1.1 can replace and improve these results. The complex Hessian equations (1.4) with χ= ωKa¨hler were solved by Dinew-Kolodziej [18], who made use of estimates of Hou [34] and Hou-MaWu [35]. For related works see [1, 17, 36, 39,”