Fully non-linear parabolic equations on compact Hermitian manifolds

A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Sz\'ekelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.

[1]  P. Eyssidieux,et al.  SINGULAR K ¨ AHLER-EINSTEIN METRICS , 2006 .

[2]  Wei Sun.  The general $J$-flows , 2015, 1507.08890.

[3]  D. Phong,et al.  Lectures on Stability and Constant Scalar Curvature , 2008, 0801.4179.

[4]  On uniform estimate of complex elliptic equations on closed Hermitian manifolds , 2014, 1412.5001.

[5]  On estimates for the Fu–Yau generalization of a Strominger system , 2015, Journal für die reine und angewandte Mathematik (Crelles Journal).

[6]  J. Spruck.  GEOMETRIC ASPECTS OF THE THEORY OF FULLY NON LINEAR ELLIPTIC EQUATIONS , 2003 .

[7]  Valentino Tosatti,et al.  The Chern–Ricci flow on complex surfaces , 2012, Compositio Mathematica.

[8]  Gábor Székelyhidi.  Fully non-linear elliptic equations on compact Hermitian manifolds , 2015, Journal of Differential Geometry.

[9]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[10]  S. Yau.  On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I* , 1978 .

[11]  Bo Guan,et al.  Complex Monge-Ampere equations and totally real submanifolds , 2009, 0910.1851.

[12]  S. Donaldson.  Moment maps and diffeomorphisms , 2002 .

[13]  S. Yau,et al.  The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation , 2006, hep-th/0604063.

[14]  Bo Guan,et al.  The dirichlet problem for a class of fully nonlinear elliptic equations , 1994 .

[15]  Xu-jia Wang,et al.  A variational theory of the Hessian equation , 2001 .

[16]  A. Zeriahi,et al.  Degenerate Complex Monge-Ampere Equations , 2017 .

[17]  Valentino Tosatti,et al.  The complex Monge-Ampère equation on compact Hermitian manifolds , 2009, 0910.1390.

[18]  Valentino Tosatti,et al.  Gauduchon metrics with prescribed volume form , 2015, 1503.04491.

[19]  S. Yau,et al.  On the parabolic kernel of the Schrödinger operator , 1986 .

[20]  S. Dinew,et al.  Liouville and Calabi-Yau type theorems for complex Hessian equations , 2012, 1203.3995.

[21]  Xiangwen Zhang,et al.  D G ] 2 7 M ay 2 01 7 THE ANOMALY FLOW ON UNIMODULAR LIE GROUPS 1 , 2017 .

[22]  Xiangwen Zhang,et al.  Anomaly flows. , 2016, 1610.02739.

[23]  Philippe Eyssidieux,et al.  Singular Kähler-Einstein metrics , 2006, math/0603431.

[24]  Huai-Dong Cao,et al.  Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds , 1985 .

[25]  Valentino Tosatti,et al.  On the evolution of a Hermitian metric by its Chern-Ricci form , 2011, 1201.0312.

[26]  X. Nie.  Regularity of A Complex Monge-Amp\`{e}re Equation on Hermitian Manifolds , 2013, 1311.4463.

[27]  N. Nguyen,et al.  Weak solutions of complex Hessian equations on compact Hermitian manifolds , 2015, Compositio Mathematica.

[28]  Zbigniew Blocki.  On uniform estimate in Calabi-Yau theorem , 2005 .

[29]  Weiling Sun.  Parabolic complex Monge–Ampère type equations on closed Hermitian manifolds , 2013, 1311.3002.

[30]  Neil S. Trudinger,et al.  Fully nonlinear, uniformly elliptic equations under natural structure conditions , 1983 .

[31]  S. Yau,et al.  A Monge?Ampère-type equation motivated by string theory , 2007 .

[32]  Regularizing properties of complex Monge–Ampère flows , 2016, 1604.06261.

[33]  Neil S. Trudinger,et al.  On the Dirichlet problem for Hessian equations , 1995 .

[34]  Valentino Tosatti,et al.  The Monge-Ampère equation for (n − 1)-plurisubharmonic functions on a compact Kähler manifold , 2013, 1305.7511.

[35]  J. Demailly,et al.  Degenerate complex Monge-Ampère equations over compact Kähler manifolds , 2010 .

[36]  Collapsing of the Chern–Ricci flow on elliptic surfaces , 2013, 1302.6545.

[37]  Hessian equations on closed Hermitian manifolds , 2015, 1501.03553.

[38]  Damin Wu,et al.  A second order estimate for complex Hessian equations on a compact Kähler manifold , 2010 .

[39]  S. Boucksom,et al.  Regularizing Properties of the Kähler–Ricci Flow , 2013 .

[40]  Taotao Zheng.  A parabolic Monge-Amp\`ere type equation of Gauduchon metrics , 2016, 1609.07854.

[41]  J. Spruck,et al.  The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian , 1985 .

[42]  M. Gill.  Convergence of the parabolic complex Monge-Amp\`ere equation on compact Hermitian manifolds , 2010, 1009.5756.

[43]  Bo Guan,et al.  SECOND-ORDER ESTIMATES AND REGULARITY FOR FULLY NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS , 2012, 1211.0181.

[44]  Xiuxiong Chen.  On the lower bound of the Mabuchi energy and its application , 2000 .

[45]  Xiangwen Zhang,et al.  Geometric flows and Strominger systems , 2015, 1508.03315.

[46]  Weiling Sun.  On a Class of Fully Nonlinear Elliptic Equations on Closed Hermitian Manifolds , 2013, 1310.0362.

[47]  L. Gårding.  An Inequality for Hyperbolic Polynomials , 1959 .

[48]  Xiangwen Zhang,et al.  2 3 M ar 2 01 8 THE ANOMALY FLOW AND THE FU-YAU EQUATION 1 , 2018 .

[49]  The Chern–Ricci Flow on Oeljeklaus–Toma Manifolds , 2017, Canadian Journal of Mathematics.

[50]  A. Hanani.  EQUATIONS DE MONGE-AMPERE SUR LES VARIETES HERMITIENNES COMPACTES , 1994 .

[51]  On estimates for fully nonlinear parabolic equations on Riemannian manifolds , 2014, 1409.3633.

[52]  T. Tô.  Regularizing properties of complex Monge–Ampère flows II: Hermitian manifolds , 2017, 1701.04023.

[53]  Convergence of the J-flow on Kähler surfaces , 2003, math/0306012.

[54]  Zbigniew Blocki.  The Complex Monge–Ampère Equation in Kähler Geometry , 2013 .

[55]  D. Phong,et al.  The Dirichlet problem for degenerate complex Monge-Ampere equations , 2009, 0904.1898.

[56]  H. Fang,et al.  On a class of fully nonlinear flows in Kähler geometry , 2009, 0904.3288.

[57]  Wei Sun.  The Parabolic Flows for Complex Quotient Equations , 2017, The Journal of Geometric Analysis.

[58]  Jian Song,et al.  D G ] 1 O ct 2 01 2 COMPLEX MONGE-AMPÈRE EQUATIONS 1 , 2014 .

[59]  Weiling Sun.  Parabolic Flow for Generalized complex Monge-Amp\`ere type equations , 2015, 1501.04255.

[60]  Xiangwen Zhang,et al.  The Fu–Yau equation with negative slope parameter , 2016, Inventiones mathematicae.

[61]  G'abor Sz'ekelyhidi,et al.  The J-flow and stability , 2013, 1309.2821.

[62]  Valentino Tosatti,et al.  Estimates for the Complex Monge-Ampère Equation on Hermitian and Balanced Manifolds , 2009, 0909.4496.

[63]  Kaising Tso,et al.  On an Aleksandrov-Bakel'Man Type Maximum Principle for Second-Order Parabolic Equations , 1985 .

[64]  Weak Solutions of the Chern-Ricci flow on compact complex surfaces , 2017, 1701.04965.

[65]  Bo Guan,et al.  On a class of fully nonlinear elliptic equations on Hermitian manifolds , 2013, Calculus of Variations and Partial Differential Equations.