Complex Monge-Amp Research Articles

Regularity of a complex Monge-Ampère equation on Hermitian manifolds

We obtain higher order estimates for a parabolic flow on a compact Hermitian manifold. As an application, we prove that a bounded $\hat{\omega}$-plurisubharmonic solution of an elliptic complex Monge-Amp\`{e}re equation is smooth under an assumption on the background Hermitian metric $\hat{\omega}$. This generalizes a result of Sz\'{e}kelyhidi and Tosatti on K\"{a}hler manifolds.

Stationary disks and Green functions in almost complex domains

Using generalized Riemann maps, normal forms for almost complex domains (D, J ) with singular foliations by stationary disks are defined. Such normal forms are used to construct counterexamples and to determine intrinsic conditions under which the stationary disks are extremal disks for the Kobayashi metric or determine solutions to almost complex Monge-Amp`ere equation.

Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors

We demonstrate how a combination of our recently developed methods of partner symmetries, symmetry reduction in group parameters and a new version of the group foliation method can produce noninvariant solutions of complex Monge-Amp\`ere equation (CMA) and provide a lift from invariant solutions of CMA satisfying Boyer-Finley equation to non-invariant ones. Applying these methods, we obtain a new noninvariant solution of CMA and the corresponding Ricci-flat anti-self-dual Einstein-K\"ahler metric with Euclidean signature without Killing vectors, together with Riemannian curvature two-forms. There are no singularities of the metric and curvature in a bounded domain if we avoid very special choices of arbitrary functions of a single variable in our solution. This metric does not describe gravitational instantons because the curvature is not concentrated in a bounded domain.

A Generalised Monge-Ampère Equation

We consider a generalised complex Monge-Amp\`ere equation on a compact K\"ahler manifold and treat it using the method of continuity. For complex surfaces, we prove an easy existence result. We also prove that (for three-folds and a related real PDE in a ball), as long as the Hessian is bounded below by a pre-determined constant (whilst moving along the method of continuity path), a smooth solution exists. Finally, we prove existence for another real PDE in a 3-ball, which is a local, real version of a conjecture of X.X. Chen.

Recursions of Symmetry Orbits and Reduction without Reduction

We consider a four-dimensional PDE possessing partner symmetries mainly on the example of complex Monge-Amp\`ere equation (CMA). We use simultaneously two pairs of symmetries related by a recursion relation, which are mutually complex conjugate for CMA. For both pairs of partner symmetries, using Lie equations, we introduce explicitly group parameters as additional variables, replacing symmetry characteristics and their complex conjugates by derivatives of the unknown with respect to group parameters. We study the resulting system of six equations in the eight-dimensional space, that includes CMA, four equations of the recursion between partner symmetries and one integrability condition of this system. We use point symmetries of this extended system for performing its symmetry reduction with respect to group parameters that facilitates solving the extended system. This procedure does not imply a reduction in the number of physical variables and hence we end up with orbits of non-invariant solutions of CMA, generated by one partner symmetry, not used in the reduction. These solutions are determined by six linear equations with constant coefficients in the five-dimensional space which are obtained by a three-dimensional Legendre transformation of the reduced extended system. We present algebraic and exponential examples of such solutions that govern Legendre-transformed Ricci-flat K\"ahler metrics with no Killing vectors. A similar procedure is briefly outlined for Husain equation.

The $\mathcal C^{2,\alpha}$ estimate of complex Monge-Ampere equation

We prove that any $\mathcal C^{1,1}$ solution to complex Monge-Amp\`ere equation $det(u_{i\bar{j}})=f$ with $0<f\in\mathcal C^{\alpha}$ is in $\mathcal C^{2,\alpha}$ for $\alpha\in (0,1)$.

Bergman metrics and geodesics in the space of Kähler metrics on toric varieties

Geodesics on the infinite dimensional symmetric space $\hcal$ of K\"ahler metrics in a fixed K\"ahler class on a projective K\"ahler manifold X are solutions of a homogeneous complex Monge-Amp\`ere equation in $X \times A$, where $A \subset \C$ is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spaces $G_{\C}/G$. Donaldson, Arezzo-Tian and Phong-Sturm raised the question whether Monge-Amp\`ere geodesics can be approximated by 1PS geodesics in the symmetric spaces of Bergman metrics. Phong-Sturm proved weak C^0 convergence of Bergman to Monge-Amp\`ere geodesics on a general \kahler manifold. In this article we prove convergence in $C^2(A \times X)$ in the case of toric K\"ahler metrics, extending our earlier result on $\CP^1$.

Intersecting solitons, amoeba, and tropical geometry

We study the generic intersection (or web) of vortices with instantons inside, which is a $1/4$ Bogomol'nyi-Prasad-Sommerfield state in the Higgs phase of five-dimensional $\mathcal{N}=1$ supersymmetric $U({N}_{\mathrm{C}})$ gauge theory on ${\mathbb{R}}_{t}\ifmmode\times\else\texttimes\fi{}({\mathbb{C}}^{*}{)}^{2}\ensuremath{\simeq}{\mathbb{R}}^{2,1}\ifmmode\times\else\texttimes\fi{}{T}^{2}$ with ${N}_{\mathrm{F}}={N}_{\mathrm{C}}$ Higgs scalars in the fundamental representation. In the case of the Abelian-Higgs model (${N}_{\mathrm{F}}={N}_{\mathrm{C}}=1$), the intersecting vortex sheets can be beautifully understood in a mathematical framework of amoeba and tropical geometry, and we propose a dictionary relating solitons and gauge theory to amoeba and tropical geometry. A projective shape of vortex sheets is described by the amoeba. Vortex charge density is uniformly distributed among vortex sheets, and negative contribution to instanton charge density is understood as the complex Monge-Amp\`ere measure with respect to a plurisubharmonic function on $({\mathbb{C}}^{*}{)}^{2}$. The Wilson loops in ${T}^{2}$ are related with derivatives of the Ronkin function. The general form of the K\"ahler potential and the asymptotic metric of the moduli space of a vortex loop are obtained as a by-product. Our discussion works generally in non-Abelian gauge theories, which suggests a non-Abelian generalization of the amoeba and tropical geometry.

Hölder continuous solutions to Monge-Ampère equations

Let $(X,\omega)$ be a compact K\"ahler manifold. We obtain uniform H\"older regularity for solutions to the complex Monge-Amp\`ere equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $\MAH(X,\omega)$ of the complex Monge-Amp\`ere operator acting on $\omega$-plurisubharmonic H\"older continuous functions. We show that this set is convex, by sharpening Ko{\l}odziej's result that measures with $L^p$-density belong to $\MAH(X,\omega)$ and proving that $\MAH(X,\omega)$ has the "$L^p$-property", $p>1$. We also describe accurately the symmetric measures it contains.

The Kähler-Einstein metric for some Hartogs domains over symmetric domains

We study the complete K\"{a}hler-Einstein metric of a Hartogs domain $\widetilde {\Omega}$, which is obtained by inflation of an irreducible bounded symmetric domain $\Omega $, using a power $N^{\mu}$ of the generic norm of $\Omega$. The generating function of the K\"{a}hler-Einstein metric satisfies a complex Monge-Amp\`{e}re equation with boundary condition. The domain $\widetilde {\Omega}$ is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by $X\in\lbrack0,1[$. This allows to reduce the Monge-Amp\`{e}re equation to an ordinary differential equation with limit condition. This equation can be explicitly solved for a special value $\mu_{0}$ of $\mu$, called the critical exponent. We work out the details for the two exceptional symmetric domains. The critical exponent seems also to be relevant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.

Partner symmetries and non-invariant solutions of four-dimensional heavenly equations

We extend our method of partner symmetries to the hyperbolic complex Monge-Amp\`ere equation and the second heavenly equation of Pleba\~nski. We show the existence of partner symmetries and derive the relations between them for both equations. For certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to systems of linear equations by an appropriate Legendre transformation. The solutions of these linear equations are generically non-invariant. As a consequence we obtain explicitly new classes of heavenly metrics without Killing vectors.

On the existence and regularity of Dirichlet problem for complex Monge-Amp�re equationson weakly pseudoconvex domains

On the existence and regularity of Dirichlet problem for complex Monge-Amp�re equationson weakly pseudoconvex domains

Partner symmetries of the complex Monge–Amp re equation yield hyper-K hler metrics without continuous symmetries

We extend the Mason-Newman Lax pair for the elliptic complex Monge-Amp\`ere equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. We identify the real and imaginary parts of the potential, which we call partner symmetries, with the translational and dilatational symmetry characteristics respectively. Then we choose the dilatational symmetry characteristic as the new unknown replacing the K\ahler potential which directly leads to a Legendre transformation and to a set of linear equations satisfied by a single real potential. This enables us to construct non-invariant solutions of the Legendre transform of the complex Monge-Amp\`ere equation and obtain hyper-K\ahler metrics with anti-self-dual Riemann curvature 2-form that admit no Killing vectors.

The general solution of the homogeneous complex Monge-Ampère equation in a space of arbitrary dimension

A general solution to the Complex Monge-Amp\`ere equation in a space of arbitrary dimensions is constructed.

A characterization of complex projective space up to biholomorphic isometry

Obata [3] gave a characterization up to isometry of the standard sphere sn in terms of the Hessian operator on a complete Riemannian manifold. With the convention Hess u = Vgrad u, his result says that if M is a complete Riemannian manifold which admits a nondegenerate function u such that Hess u = -u- Id then M is isometric to the standard sphere. Obata goes on to prove related results in conformal geometry which take advantage of the existence of a function whose Hessian has a special form. Other authors have also obtained strong geometric properties of a Riemannian manifold by exploiting the existence of a function u with Hess u = f - Id for some function f. In particular, this last equation implies that M is a warped product. For a proof and related results, see, for example, Osgood and Stowe [5]. In the complex case, a characterization of cn up to isometry was given by Stoll [6] via the complex Monge-Amp~re operator. Stoll's result says that if M is a complex manifold which admits a strictly plurisubharmonic exhaustion r: M --> [0, ~) such that (dd c log r) n -- 0, then (M, r) = (C", Iz12). In other words M, with hermitian metric given by the Kahler form ddCr, is biholomorphically isometric to C". Obata also showed that a complete, connected and simply connected Kahler manifold is isometric to the complex projective space ]?n if and only if it admits a solution to a certain linear system of third order differential equations [4]. Blair [1] subsequently showed that in some cases this characterization of ]?n follows from a corresponding result for Riemannian manifolds and indicated that one would not expect a characterization of ]?n by a Hessian equation analogous to that which Obata used to characterize S n. In this paper we give a complex analog of Obata's theorem [3]. We characterize complex projective space up to biholomorphic isometry by the existence of a solution to a system of second order equations. Since ]?n with the Fubini-Study metric is not a warped product, there does not exist a nontrivial function u on ]?n whose Hessian is a multiple of the identity. However, ]?n with one point deleted is the hyperplane section bundle over I? n- 1 and the fibers of this bundle are totally geodesic complex lines. Thus, there is a relationship between the natural metric structure and the line bundle structure of ~n. This relationship provides the motivation for the construction of a function u on ]?n

Sums of continuous plurisubharmonic functions and the complex Monge-Amp�re operator in ? n

Sums of continuous plurisubharmonic functions and the complex Monge-Amp�re operator in ? n

Solving the degenerate complex Monge-Amp�re equation with one concentrated singularity

Solving the degenerate complex Monge-Amp�re equation with one concentrated singularity

The dirichlet problem for a complex Monge-Amp�re equation

where G' denotes the Jacobian determinant of G. Furthermore, if G4=(0 . . . . . 0), then (d d ~ log h G I) 0. Thus, for f = 0, (1) is a natural generalization of the Dirichlet problem for harmonic in the complex plane. Other extended Dirichlet problems were studied in connection with function theory in several variables by S. Bergman [2, 3] (on domains with distinguished boundary surfaces) and more generally by H. Bremermann [4]. In Section 8 it is shown that the solution of the problem discussed by Bremermann actually solves (1), in a generalized sense, with f = 0 . The problem (1) seems to be a reasonable candidate for a (nonlinear) potential theory associated with the theory of of several complex variables. The question of uniqueness for the problem (1) is related to the question of existence of inner functions on the domain O. If h is a bounded analytic function