We completely classify three-dimensional Lorentz manifolds, curvature homogeneous up to order one, equipped with Einstein-like metrics. New examples arise with respect to both homogeneous examples and three-dimensional Lorentz manifolds admitting a degenerate parallel null line field.

We prove that the least perimeter $P(n)$ of a partition of a smooth, compact Riemannian surface into $n$ regions of equal area $A$ is asymptotic to $n/2$ times the perimeter of a planar regular hexagon of area $A$. Along the way, we derive tighter estimates for flat tori, Klein bottles, truncated cylinders, and Möbius bands.

In this paper, we define and study the geometry of semi-slant submanifolds of an almost paracontact metric manifold. We give some characterizations for a submanifold to be semi-slant submanifold to be semi-slant product and obtain integrability conditions for the distributions involved in the definition of a semi-slant submanifold.

On a compact K\"{a}hler manifold $X$ with a holomorphic 2-form $\a$, there is an almost complex structure associated with $\a$. We show how this implies vanishing theorems for the Gromov--Witten invariants of $X$. This extends the approach used by Parker and the author for K\"{a}hler surfaces to higher dimensions.

The solutions to Hilbert's Fourth Problem in the regular case are projectively flat Finsler metrics. In this paper, we consider the so-called $(\alpha,\beta)$-metrics defined by a Riemannian metric $\alpha$ and a $1$-form $\beta$, and find a necessary and sufficient condition for such metrics to be projectively flat in dimension $n \geq 3$.

In this paper we study holomorphic maps between almost Hermitian manifolds. We obtain a new criterion for the harmonicity of such holomorphic maps, and we deduce some applications to horizontally conformal holomorphic submersions.

Biharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed.

In this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact K\"ahler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type $(A)$ in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator.

Let $M$ be a symplectic $4$-dimensional manifold equipped with a Hamiltonian circle action with isolated fixed points. We describe a method for computing its integral equivariant cohomology in terms of fixed point data. We give some examples of these computations.

Recently I. Castro and F. Urbano introduced the Lagrangian catenoid. Topologically, it is $\mathbb R\times S^{n-1}$ and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ${\mathbb C}^n$ is foliated by round $(n-1)$-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ${\mathbb C}^n$. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid.

We prove the non existence of real hypersurfaces in complex projective space whose structure Jacobi operator is of Codazzi type.

Let $M$ be an $m$ dimensional submanifold in the Euclidean space ${\mathbf R}^n$ and $H$ be the mean curvature of $M$. We obtain some low geometric estimates of the total square mean curvature $\int_M H^2 d\sigma$. The low bounds are geometric invariants involving the volume of $M$, the total scalar curvature of $M$, the Euler characteristic and the circumscribed ball of $M$.

In this paper, we consider Hermitian harmonic maps from Hermitian manifolds into convex balls. We prove that there exist no non-trivial Hermitian harmonic maps from closed Hermitian manifolds into convex balls, and we use the heat flow method to solve the Dirichlet problem for Hermitian harmonic maps when the domain is a compact Hermitian manifold with non-empty boundary.

We study a real hypersurface $M$ in a complex space form $\mn$, $c \neq 0$, whose shape operator and structure tensor commute each other on the holomorphic distribution of $M$.

We classify the left-invariant metrics with nonnegative sectional curvature on $\SO(3)$ and $U(2)$.

We show that given a stable weighted configuration on the asymptotic boundary of a locally compact Hadamard space, there is a polygon with Gauss map prescribed by the given weighted configuration. Moreover, the same result holds for semistable configurations on arbitrary Euclidean buildings.

A sharp upper bound on the first $S^{1}$ invariant eigenvalue of the Laplacian for $S^1$ invariant metrics on $S^2$ is used to find obstructions to the existence of $S^1$ equivariant isometric embeddings of such metrics in $(\R^3,\can)$. As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This leads to generalizations of some classical results in the theory of surfaces.

In this paper we give a characterization of real hypersurfaces of type $A$ in a complex two-plane Grassmannian $G_2(\mathbb{C}^{m+2})$ which are tubes over totally geodesic $G_2(\mathbb{C}^{m+1})$ in $G_2(\mathbb{C}^{m+2})$ in terms of the {\it vanishing Lie derivative\/} of the shape operator $A$ along the direction of the Reeb vector field $\xi$.

We investigate the geometry of manifolds with bounded Ricci curvature in $L^1$-sense. In particular, we generalize the classical volume comparison theorem to our situation and obtain a generalized sphere theorem.

Using a modification of Webster's proof of the Newlander--Nirenberg theorem, it is shown that, for a weakly convergent sequence of integrable unitary connections on a complex vector bundle over a complex manifold, there is a subsequence of local holomorphic frames that converges strongly in an appropriate Holder class.

We study Hamiltonian actions of compact groups in the presence of compatible involutions. We show that the Lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our techniques imply that the solution to the eigenvalues of a sum problem for a given real form can be reduced to the quasi-split real form in the same inner class. We also consider invariant quotients with respect to the corresponding real form of the complexified group.

In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension $n \geq 3$. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler metric is locally projectively flat

In this paper, we consider Yang-Mills connections on a vector bundle $E$ over a compact Riemannian manifold $M$ of dimension $m> 4$, and we show that any set of Yang-Mills connections with the uniformly bounded $L^{\frac{m}{2}}$-norm of curvature is compact in $C^{\infty}$ topology.

The main purpose of this paper is to determine isotropic immersions of complex space forms into real space forms with low codimension. This is an improvement of a result of S. Maeda.