On holomorphic harmonic morphisms

We study holomorphic harmonic morphisms from K?hler manifolds to almost Hermitian manifolds. When the codomain is also K?hler we get restrictions on such maps in the case of constant holomorphic curvature. We also prove a Bochner-type formula for holomorphic harmonic morphisms which, under certain curvature conditions of the domain, gives insight to the structure of the vertical distribution. We thus prove that when the domain is compact and non-negatively curved, the vertical distribution is totally geodesic. Received: 28 May 2001     

Non-holomorphic harmonic morphisms from Kähler manifolds

We give a method for constructing non-holomorphic harmonic morphisms from Kähler manifolds. The method is then used to obtain many such examples defined locally on the complex Grassmannians.     

Minimal submanifolds of hyperbolic spaces via harmonic morphisms

In this paper we give a method for constructing complete minimal submanifolds of the hyperbolic spaces H ^m . They are regular fibres of harmonic morphisms from H ^m with values in Riemann surfaces.     

Finite time blow-up for the harmonic map heat flow

Summary We consider the harmonic map heat flow from the three-dimensional ball to the two-sphere. We establish the existence of regular initial data leading to blow-up in finite time.     

The Hessian and Jacobi morphisms for higher order calculus of variations

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second variation is equal (up to horizontal differentials) to the vertical differential of the Euler-Lagrange morphism which turns out to be self-adjoint along solutions of the Euler-Lagrange equations. These two objects, respectively, generalize in an invariant way the Hessian morphism and the Jacobi morphism (which is then self-adjoint along critical sections) of a given Lagrangian to the case of higher order Lagrangians. Some examples of classical Lagrangians are provided to illustrate our method.     

Symmetric boundary values for the Dirichlet problem for harmonic maps from the disc into the 2-sphere

Let Aut(D) denote the group of biholomorphic diffeormorphisms from the unit disc D onto itself and O(3) the group of orthogonal transformations of the unit sphere S ². The existence of multiple solutions to the Dirichlet problem for harmonic maps from D into S ² is related to the symmetries (if any) of the boundary value γ : ∂D → S ², by invariance of the Dirichlet energy under the action of Aut(D) × O(3). In this paper, we classify the stabilizers in Aut(D) × O(3) of boundary values in H ^1/2(S ¹, S ²) and . We give two applications to the Dirichlet problem for harmonic maps. This work was partially supported by the CMLA, Ecole Normale Supérieure de Cachan, Cachan, France.     

A free convenient vector space for holomorphic spaces

In this paper we show that there exists a free convenient vector space for the case of holomorphic spaces and holomorphic maps. This means that for every spaceX with a holomorphic structure, there exists an appropriately complete locally convex vector space X and a holomorphic mapl _X:XX, such that for any vector space of the same kind the map (l _X )^*:L(X,E)(X,E) is a bijection. Analogously to the smooth case treated in [2, 5.1.1] the free convenient vector space X can be obtained as the Mackey closure of the linear subspace spanned by the image of the canonical mapX(X).In the second part of the paper we prove that in the case whereX is a Riemann surface, one hasX=(X,).     

Composite of irreducible morphisms in the bounded derived category

We study necessary and sufficient conditions for the existence of n irreducible morphisms in the bounded derived category of an Artin algebra, with non-zero composite in the n+1-power of the radical. In the case of , the bounded derived category of an Ext-finite hereditary k-category with tilting object, such irreducible morphisms exist if and only if H is derived equivalent to a wild hereditary algebra or to a wild canonical algebra. We also characterize the cluster tilted algebras having such irreducible morphisms.     

A note on effective descent morphisms of topological spaces and relational algebras

We formulate two open problems related to and, in a sense, suggested by the Reiterman-Tholen characterization of effective descent morphisms of topological spaces.     

Degrees of irreducible morphisms in standard components

We study the left degree of an irreducible morphism with X and Y_i indecomposable modules in a standard component of the Auslander-Reiten quiver, for 1≤i≤r. Two criteria to determine whether the left degree of these irreducible morphisms is finite or infinite are given, for standard algebras. We also study which of them has left degree two.     

On the composite of irreducible morphisms in almost sectional paths

We study here when the composite of n irreducible morphisms in almost sectional paths is non-zero and lies in ℜⁿ⁺¹.     

A Dirichlet problem for harmonic forms inC 1-domains

A Dirichlet problem for harmonic forms in openC ¹-domains is here investigated with boundary data of classL ^p .Work partially supported by MURST 60% and 40%.     

Galois closure of essentially finite morphisms

Let X be a reduced connected k-scheme pointed at a rational point x∈X(k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphism f:Y→X satisfying the condition H⁰(Y,O_Y)=k; this Galois closure is a torsor dominating f by an X-morphism and universal for this property. Moreover, we show that is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X. We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f:Y→X under a finite group scheme satisfying the condition H⁰(Y,O_Y)=k, Y has a fundamental group scheme π₁(Y,y) fitting in a short exact sequence with π₁(X,x).     

On holomorphic Banach vector bundles over Banach spaces

Let X be a Banach space with a countable unconditional basis (e.g., X = ℓ ₂ Hilbert space), Ω ⊂ X pseudoconvex open, E → Ω a locally trivial holomorphic vector bundle with a Banach space Z for fiber type, the sheaf of germs of holomorphic sections of E → Ω, and Z ₁ the Banach space . Then and Ω × Z ₁ are holomorphically isomorphic, is acyclic and E is so to speak stably trivial over Ω in a generalized sense. We also show that if E is continuously trivial over Ω, then E is holomorphically trivial over Ω. In particular, if Z = ℓ ₂ or Ω is contractible, then E is holomorphically trivial over Ω. Some applications are also given. To my dear little daughter, Sári Mangala, on her third birthday. Supported in part by a Research Initiation Grant from Georgia State University.     

Liouville theorem for harmonic maps with potential

LetM, N be complete manifolds,u:M →N be a harmonic map with potentialH, namely, a critical point of the functionalE _H (u)=∫ _M [e(u) − H(u)], wheree(u) is the energy density ofu. We will give a Liouville theorem foru with a class of potentialsH’s. Research supported in part by NNSFC, SFECC and NSFCCNU.     

Extremal holomorphic diffusion processes

Extremal holomorphic diffusion processes are studied. We formulate a stochastic control problem for the extremal function of a set. We then characterize the extremal holomorphic diffusion processes as the optimal diffusion processes of the problem. By making use of SDE representation for the processes, we show that they move on an integral submanifold of the coefficients vector fields of the SDE passing through the starting point.     

Liouville theorems for exponentially harmonic functions on Riemannian manifolds

Suppose thatM is a complete Riemannian manifolds with nonnegative sectional curvature. We prove that any bounded exponentially harmonic function onM is a constant function.     

Constructions of harmonic polynomial maps between spheres

The complexity of _q -eigenmaps, i.e. homogeneous degreeq harmonic polynomial mapsf:S ^m S ⁿ, increases fast with the degreeq and the source dimensionm. Here we introduce a variety of methods of manufacturing new eigenmaps out of old ones. They include degree and source dimension raising operators. As a byproduct, we get estimates on the possible range dimensions of full eigenmaps and obtain a geometric insight of the harmonic product of ₂-eigenmaps.