We associate with a Kähler manifold (E2n,J,ω) equipped with a Killing holomorphic field Z with non vanishing Hamiltonian f, ωºZ=df, a conical Kähler manifold Ê2n+2, called conification of E.
It is used for construction a 1-parametric family of conical hyper-Kähler manifolds Ê4n+4 associated with a hyper-Kähler manifold (E4n,g,Jα,ωα) equipped with a Killing vector field Z which preserves the complex structure J1, interchanges J2 and J3 and has non-vanishing ω1-Hamiltonian.
We show that this construction can be applied to the cotangent bundle N4n+4=T*Ê2n+2 of a special conical Kähler manifold Ê2n+2 with natural hyper-Kähler structure.
This gives a purely differential geometric description and generalization of supergravity c-map which associates with a projective special Kähler manifold M2n=Ê2n+2/C* a quaternionic Kähler manifold N4n+4/(H*/Z2).
Joint work with V. Cortes, M. Dyckmanns and Th. Mohaupt.
I will describe a new gluing result for Kähler constant scalar curvature metrics obtained in collaboration with R. Lena and L. Mazzieri. This construction generalises to Kcsc metrics of any sign the celebrated Kummer construction for Calabi-Yau manifolds discovered by Page, Topiwala and LeBrun and Singer.
I shall describe a new construction of quaternionic Kähler metrics in dimension 8 from the G2 geometry of a generic plane distribution in dimension 5. I will explain how this construction fits in a general program of R. Mazzeo and I on constructing Einstein metrics by a nonlinear Poisson transform.
CR structures in codimension one play an increasingly important role in differential geometry, deeply intertwined with Kähler geometry. In this talk, based on joint work in progress with V. Apostolov, P. Gauduchon and E. Legendre, I discuss the relation between CR structures in higher codimension and Kähler geometry, through a process called "Levi-Kähler reduction". I focus in particular on the toric case, where Levi-Kähler reduction provides a new way to construct distinguished metrics on toric varieties. When the Delzant polytope is a product of simplices, explicit quotients of products of spheres are obtained, generalizing Bryant's Bochner-flat metrics on weighted projective spaces.
It is shown that compact homogeneous locally conformally Kähler manifolds are Vaisman (joint work with Andrei Moroianu and Liviu Ornea).
I'll discuss in which sense general metric measure structures possess a (co)tangent bundle and how to assign to Sobolev functions a differential and a gradient. If time permits, I'll also show how in spaces with a lower bound on the Ricci curvature this line of thought brings to the notions of Hessian, covariant and exterior differentiation and to the definition of the Ricci curvature tensor.
Let M be the underlying 4-manifold of a Del Pezzo surface. We show that a specific open region in the space of Riemannian metrics on M contains all the known Einstein metrics on M, but no others; consequently, this region contributes exactly one connected component to the moduli space of Einstein metrics on M. Our methods also yield new results concerning the geometry of almost-Kähler 4-manifolds.
I will present and discuss some results and problems about some flows of metrics on Riemannian manifolds correlated to Ricci flow:
- The "renormalization group" flow, truncated at the second order term. The Ricci flow is its trucation at the first order (joint work with L. Cremaschi).
- The "Ricci-Bourguignon" flow, which is a perturbation of the Ricci flow equation by an extra term proportional to the product of the scalar curvature with the metric tensor (joint work with G. Catino, L. Cremaschi, Z. Djadli, L. Mazzieri).
- A "noname" flow that I and N. Gigli introduced using the theory of optimal transport of mass, which is "tangent" to the Ricci flow at the initial time and which can be defined also for nonsmooth metric spaces.
Given a planar domain Ω and a function f(t) we present some results related with the classical question of Berestycki, Caffarelli and Nirenberg on bounded positive solutions of the overdetermined problem
Δu + f(u) = 0 in Ω
u = 0 and ∂u/∂n = C on ∂Ω
The work of Colding-Minicozzi gives that a sequence of embedded minimal disks converges, up to a subsequence, to a minimal lamination away from a closed set of singular points. In several examples of Colding-Minicozzi and others, the leaves of such lamination are disks, while Hoffman-White recently produced examples where some of the leaves are annuli. In this talk I will describe several results on the topology of the leaves of such lamination in a manifold that admits an isoperimetric inequality for minimal surfaces. For instance, each leaf has genus zero. This is joint work with Bernstein.
We associate with a Kähler manifold (E2n,J,ω) equipped with a Killing holomorphic field Z with non vanishing Hamiltonian f, ωºZ=df, a conical Kähler manifold Ê2n+2, called conification of E.
caIt is used for construction a 1-parametric family of conical hyper-Kähler manifolds Ê4n+4 associated with a hyper-Kähler manifold (E4n,g,Jα,ωα) equipped with a Killing vector field Z which preserves the complex structure J1, interchanges J2 and J3 and has non-vanishing ω1-Hamiltonian.
We show that this construction can be applied to the cotangent bundle N4n+4=T*Ê2n+2 of a special conical Kähler manifold Ê2n+2 with natural hyper-Kähler structure.
This gives a purely differential geometric description and generalization of supergravity c-map which associates with a projective special Kähler manifold M2n=Ê2n+2/C* a quaternionic Kähler manifold N4n+4/(H*/Z2).
Joint work with V. Cortes, M. Dyckmanns and Th. Mohaupt.
I will describe a new gluing result for Kähler constant scalar curvature metrics obtained in collaboration with R. Lena and L. Mazzieri. This construction generalises to Kcsc metrics of any sign the celebrated Kummer construction for Calabi-Yau manifolds discovered by Page, Topiwala and LeBrun and Singer.
I shall describe a new construction of quaternionic Kähler metrics in dimension 8 from the G2 geometry of a generic plane distribution in dimension 5. I will explain how this construction fits in a general program of R. Mazzeo and I on constructing Einstein metrics by a nonlinear Poisson transform.
CR structures in codimension one play an increasingly important role in differential geometry, deeply intertwined with Kähler geometry. In this talk, based on joint work in progress with V. Apostolov, P. Gauduchon and E. Legendre, I discuss the relation between CR structures in higher codimension and Kähler geometry, through a process called "Levi-Kähler reduction". I focus in particular on the toric case, where Levi-Kähler reduction provides a new way to construct distinguished metrics on toric varieties. When the Delzant polytope is a product of simplices, explicit quotients of products of spheres are obtained, generalizing Bryant's Bochner-flat metrics on weighted projective spaces.
It is shown that compact homogeneous locally conformally Kähler manifolds are Vaisman (joint work with Andrei Moroianu and Liviu Ornea).
I'll discuss in which sense general metric measure structures possess a (co)tangent bundle and how to assign to Sobolev functions a differential and a gradient. If time permits, I'll also show how in spaces with a lower bound on the Ricci curvature this line of thought brings to the notions of Hessian, covariant and exterior differentiation and to the definition of the Ricci curvature tensor.
Let M be the underlying 4-manifold of a Del Pezzo surface. We show that a specific open region in the space of Riemannian metrics on M contains all the known Einstein metrics on M, but no others; consequently, this region contributes exactly one connected component to the moduli space of Einstein metrics on M. Our methods also yield new results concerning the geometry of almost-Kähler 4-manifolds.
I will present and discuss some results and problems about some flows of metrics on Riemannian manifolds correlated to Ricci flow:
- The "renormalization group" flow, truncated at the second order term. The Ricci flow is its trucation at the first order (joint work with L. Cremaschi).
- The "Ricci-Bourguignon" flow, which is a perturbation of the Ricci flow equation by an extra term proportional to the product of the scalar curvature with the metric tensor (joint work with G. Catino, L. Cremaschi, Z. Djadli, L. Mazzieri).
- A "noname" flow that I and N. Gigli introduced using the theory of optimal transport of mass, which is "tangent" to the Ricci flow at the initial time and which can be defined also for nonsmooth metric spaces.
Given a planar domain Ω and a function f(t) we present some results related with the classical question of Berestycki, Caffarelli and Nirenberg on bounded positive solutions of the overdetermined problem
Δu + f(u) = 0 in Ω
u = 0 and ∂u/∂n = C on ∂Ω
The work of Colding-Minicozzi gives that a sequence of embedded minimal disks converges, up to a subsequence, to a minimal lamination away from a closed set of singular points. In several examples of Colding-Minicozzi and others, the leaves of such lamination are disks, while Hoffman-White recently produced examples where some of the leaves are annuli. In this talk I will describe several results on the topology of the leaves of such lamination in a manifold that admits an isoperimetric inequality for minimal surfaces. For instance, each leaf has genus zero. This is joint work with Bernstein.