We consider dimension reduction for solutions of the Kähler-Ricci flow with nonegative bisectional curvature. When the complex dimension $n=2$, we prove an optimal dimension reduction theorem for complete translating Kähler-Ricci solitons with nonnegative bisectional curvature. We also prove a general dimension reduction theorem for complete ancient solutions of the Kähler-Ricci flow with nonnegative bisectional curvature on noncompact complex manifolds under a finiteness assumption on the Chern number $c^n_1$.
Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A crucial step in the proof is to show a vanishing theorem of Dolbeault-Morse-Novikov cohomology.
By establishing two general quadratic inequalities, we obtain some inequalities related to Ricci curvatures for Lagrangian submanifolds of K$\ddot{\mathrm{a}}$hler QCH-manifolds, which generalize some results for Lagrangian submanifolds of complex space forms.
We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log terminal singularities.
Let $(M^3,g_0)$ be a complete noncompact Riemannian 3-manifold with nonnegative Ricci curvature and with injectivity radius bounded away from zero. Suppose that the scalar curvature $R(x)\to 0$ as $x\to \infty$. Then the Ricci flow with initial data $(M^3,g_0)$ has a long time solution. This extends a recent result of Ma and Zhu. We also have a higher dimensional version, and we reprove a K$\ddot{a}$hler analogy due to Chau, Tam and Yu.
For any complete noncompact K$\ddot{a}$hler manifold with nonnegative and bounded holomorphic bisectional curvature,we provide the necessary and sufficient condition for non-ancient solution to the Ricci flow in this paper.
We revisit generalized K$\ddot{a}$hler reduction introduced by Lin and Tolman in \cite{LT} from a viewpoint of geometric invariant theory. It is shown that in the strong Hamiltonian case introduced in the present paper, many well-known conclusions of ordinary K$\ddot{a}$hler reduction can be generalized without much effort to the generalized setting. It is also shown how generalized holomorphic structures arise naturally from the reduction procedure.
We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.
We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K $\in$ R on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2$π$ and its dimension is at most equal to N. This gives a new wide class of geometric examples of metric measure spaces satisfying the RCD(K, N) curvature-dimension condition, including for instance spherical suspensions, orbifolds, K{ä}hler-Einstein manifolds with a divisor, Einstein manifolds with conical singularities along a curv…
We study pluripotential complex Monge-Ampère flows in big cohomology classes on compact K{ä}hler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential K{ä}hler-Ricci flows on compact K{ä}hler manifolds of general type as well as on stable varieties.
In this paper, metric reduction in generalized geometry is investigated. We show how the Bismut connections on the quotient manifold are obtained from those on the original manifold. The result facilitates the analysis of generalized K$\ddot{a}$hler reduction, which motivates the concept of metric generalized principal bundles and our approach to construct a family of generalized holomorphic line bundles over $\mathbb{C}P^2$ equipped with some non-trivial generalized K$\ddot{a}$hler structures.
If g(t) is a three-dimensional Ricci flow solution, with sectional curvatures that decay like the inverse of t and diameter that increases at most like the square root of t, then the pullback Ricci flow solution on the universal cover approaches a homogeneous expanding soliton.
We consider gradient Ricci solitons conformal to a $n$-dimensional pseudo-Euclidean space and we completely describe the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations. As a consequence, the gradient Ricci solitons that arise from the reduced system are invariant under the action of either an $(n-1)$-dimensional translation group or the pseudo-orthogonal group acting on the corresponding $n$-dimensional pseudo-Euclidean space.
This paper is devoted to obtain the one-dimensional group invariant solutions of the two-dimensional Ricci flow ((2D) Rf) equation. By classifying the orbits of the adjoint representation of the symmetry group on its Lie algebra, the optimal system of one-dimensional subalgebras of the ((2D) Rf) equation is obtained. For each class, we will find the reduced equation by the method of similarity reduction. By solving these reduced equations, we will obtain new sets of group invariant solutions for the ((2D) Rf) equation.
We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural assumptions on the Cauchy-Dirichlet boundary data, we show that the envelope of pluripotential subsolutions is semi-concave in time and continuous in space, and provides the unique pluripotential solution with such regularity.
Let $M^n$ be a complete noncompact K$\ddot{a}$hler manifold of complex dimension $n$ with nonnegative holomorphic bisectional curvature. Denote by $\mathcal{O}$$_d(M^n)$ the space of holomorphic functions of polynomial growth of degree at most $d$ on $M^n$. In this paper we prove that $$dim_{\mathbb{C}}{\mathcal{O}}_d(M^n)\leq dim_{\mathbb{C}}{\mathcal{O}}_{[d]}(\mathbb{C}^n),$$ for all $d>0$, with equality for some positive integer $d$ if and only if $M^n$ is holomorphically isometric to $\mathbb{C}^n$. We also obtain sharp improved dimension estimates when its volume growth is not maximal or…
Given the Euclidean space $\R^{2n+2}$ endowed with a constant symplectic structure and the standard flat connection, and given a polynomial of degree 2 on that space, Baguis and Cahen have defined a reduction procedure which yields a symplectic manifold endowed with a Ricci-type connection. We observe that any symplectic manifold of dimension greater than 2 endowed with a symplectic connection of Ricci-type is locally given by a local version of such a reduction. We also consider the reverse of this reduction procedure, an induction procedure: we construct globally on a symplectic manifold end…
We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi--Yau geometry and analysis and the Kodaira--Kuranishi--Spencer theory and building up on results of N.Koiso for the case of compact manifolds, we show that under rather general hypotheses any `small' asymptotically cylindrical Ricci-flat deformations of asymptotically cylindrical Ricci-flat Kähler metrics are again Kähler, possibly with respect to a perturbed complex structure. We also find the dimension of the moduli space for these small deformations. In the class o…
In this short note, we prove that the only simply connected noncompact three-dimensional Type I $κ$-solution to the Ricci flow is the shrinking cylinder. This work can be regarded as a generalization of Cao and Chow, and a complement of Ding and Ni. Up to this point, three-dimensional $κ$-solutions of Type I are completely classified, and it remains interesting to work further towards Perelman's assertion, that the only remaining possibility of three-dimensional noncompact $κ$-solution is the Bryant soliton. Brendle is working to that end. The classification of three-dimensional $κ$-solution i…
Let $M$ be a Fano manifold equipped with a Kähler form $ω\in 2πc_1(M)$ and $K$ a connected compact Lie group acting on $M$ as holomorphic isometries. In this paper, we show the minimality of a $K$-invariant Lagrangian submanifold $L$ in $M$ w.r.t. a globally conformal Kähler metric is equivalent to the minimality of the reduced Lagrangian submanifold $L_0=L/K$ in a Kähler quotient $M_0$ w.r.t. the Hsiang-Lawson metric. Furthermore, we give some examples of Kähler reductions by using a circle action obtained from a cohomogenenity one action on a Kähler-Einstein manifold of positive Ricci curvat…
Under broad hypotheses we derive a scalar reduction of the generalized Kähler-Ricci soliton system. We realize solutions as critical points of a functional analogous to the classical Aubin energy defined on the orbit of a natural Hamiltonian action of diffeomorphisms, thought of as a generalized Kähler class. This functional is convex on a large set of paths in this space, and using this we show rigidity of solitons in their generalized Kähler class. As an application we prove uniqueness of the generalized Kähler-Ricci solitons on Hopf surfaces constructed in arXiv:1907.03819, finishing the cl…
In this paper, we first show an interpretation of the Kähler-Ricci flow on a manifold $X$ as an exact elliptic equation of Einstein type on a manifold $M$ of which $X$ is one of the (Kähler) symplectic reductions via a (non-trivial) torus action. There are plenty of such manifolds (e.g. any line bundle on $X$ will do). Such an equation is called $V$-soliton equation, which can be regarded as a generalization of Kähler-Einstein equations or Kähler-Ricci solitons. As in the case of Kähler-Einstein metrics, we can also reduce the $V$-soliton equation to a scalar equation on Kähler potentials, whi…
We investigate how to obtain various flows of Kähler metrics on a fixed manifold as variations of Kähler reductions of a metric satisfying a given static equation on a higher dimensional manifold. We identify static equations that induce the geodesic equation for the Mabuchi's metric, the Calabi flow, the pseudo-Calabi flow of Chen-Zheng and the Kähler-Ricci flow. In the latter case we re-derive the V-soliton equation of La Nave-Tian.
Ansatze are constructed under which the solutions of 11D supergravity must be stationary points of a parabolic flow on a Riemannian manifold $M^{10-p}$. This parabolic flow turns out to be the Ricci flow coupled to a scalar field, a $(3-p)$-form, and a $4$-form. This allows the introduction of techniques from parabolic partial differential equations to the search of solutions to 11D supergravity. As a first step, Shi-type estimates and criteria for the long-time existence of the flow are established.
