Complex Hessian Equations on Some Compact Kähler Manifolds
2012 ◽
Vol 2012
◽
pp. 1-48
◽
Keyword(s):
A Priori
◽
On a compact connected2m-dimensional Kähler manifold with Kähler formω, given a smooth functionf:M→ℝand an integer1<k<m, we want to solve uniquely in[ω]the equationω̃k∧ωm-k=efωm, relying on the notion ofk-positivity forω̃∈[ω](the extreme cases are solved:k=mby (Yau in 1978), andk=1trivially). We solve by the continuity method the corresponding complex elliptickth Hessian equation, more difficult to solve than the Calabi-Yau equation (k=m), under the assumption that the holomorphic bisectional curvature of the manifold is nonnegative, required here only to derive an a priori eigenvalues pinching.
1976 ◽
Vol 12
(1)
◽
pp. 191-214
◽
1976 ◽
Vol 12
(2)
◽
pp. 439-445
◽
1980 ◽
pp. 412-428
◽
1990 ◽
Vol 23
(2)
◽
pp. 437-441
◽
2013 ◽
Vol 24
(3)
◽
pp. 1583-1612
◽
1986 ◽
pp. 90-103
◽
1975 ◽
pp. 113-123
◽