The Yamabe problem for almost Hermitian manifolds

The conformal class of a Hermitian metric g on a compact almost complex manifold (M^2m, J) consists entirely of metrics that are Hermitian with respect to J. For each one of these metrics, we may define a J-twisted version of the Ricci curvature, the J-Ricci curvature, and its corresponding trace, the J-scalar curvature s^J. We ask if the conformal class of g carries a metric with constant s^J, an almost Hermitian version of the usual Yamabe problem posed for the scalar curvature s. We answer our question in the affirmative. In fact, we show that (2m−1)s^J−s=2(2m−1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g. Using techniques developed for the solution of the problem for s, we construct an almost Hermitian Yamabe functional and its corresponding conformal invariant. This invariant is bounded from above by a constant that only depends on the dimension of M, and when it is strictly less than the universal bound, the problem has a solution that minimizes the almost complex Yamabe functional. By the relation above, we see that when W (ω, ω) is negative at least one point, or identically zero, our problem has a solution that minimizes the almost Hermitian Yamabe functional, and the universal bound is reached only in the case of the standard 6-sphere equipped with a suitable almost complex structure. When W(ω, ω) is non-negative and not identically zero, we prove that the conformal invariant is strictly less than the universal bound, thus solving the problem for this type of manifolds as well. We discuss some applications.     

Prescribing scalar and boundary mean curvature on the three dimensional half sphere

We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half-three sphere, by deforming conformally its standard metric. Using blow-up analysis techniques and minimax arguments, we prove some existence and compactness results.     

On the negative case of the Singular Yamabe Problem

Let (M, g) be a compact Riemannian manifold of dimension n ≥3, and let Γ be a nonempty closed subset of M. The negative case of the Singular Yamabe Problem concerns the existence and behavior of a complete metric g on M∖Γ that has constant negative scalar curvature and is pointwise conformally related to the smooth metric g. Previous results have shown that when Γ is a smooth submanifold (with or without boundary) of dimension d, there exists such a metric if and only if . In this paper, we consider a more general class of closed sets and show the existence of a complete conformai metric ĝ with constant negative scalar curvature which depends on the dimension of the tangent cone to Γ at every point. Specifically, provided Γ admits a nice tangent cone at p, we show that when the dimension of the tangent cone to Γ at p is less than then there cannot exist a negative Singular Yamabe metric ĝ on M∖Γ.     

Local volume estimate for manifolds with<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-bounded curvature

We obtain a local volume growth for complete, noncompact Riemannian manifolds with small integral bounds and with Bach tensor having finite L² norm in dimension 4.     

The isoperimetric problem in complete surfaces of nonnegative curvature

We show that in a complete plane with nonnegative curvature there is a perimeter minimizing set of any given area. This set is a disc whose boundary is a closed embedded curve with constant geodesic curvature.     

The convex-concave principle and uniqueness for hypersurfaces in space Forms

We establish very general Weyl identities for pairs of symmetric functions of the invariants of the shape operator of a hypersurface in a space form or a general Codazzi tensor, respectively. They are used to characterize certain isoparametric hypersurfaces by assuming constancy or extremal properties of the functions, provided they fulfill ellipticity and/or convexity (concavity) properties. This way, many wellknown results are generalized. Finally, a chain rule for Weyl identities offers additional extension of some results.     

The Weyl functional near the Yamabe invariant

For a compact manifold M ofdim M=n≥4, we study two conformal invariants of a conformal class C on M. These are the Yamabe constant Y_C(M) and the L^n/2-norm W_C(M) of the Weyl curvature. We prove that for any manifold M there exists a conformal class C such that the Yamabe constant Y_C(M) is arbitrarily close to the Yamabe invariant Y(M), and, at the same time, the constant W_C(M) is arbitrarily large. We study the image of the mapYW:C→(Y_C(M), W_C(M))∈R ² near the line {(Y(M), w)|w∈R}. We also apply our results to certain classes of 4-manifolds, in particular, minimal compact Kähler surfaces of Kodaira dimension 0, 1 or 2.     

Existence and uniqueness of complete constant mean curvature surfaces at infinity ofH 3

A set of conditions are given, each equivalent to the constancy of mean curvature of a surface in H ³.It is shown that analogs of these equivalences exist for surfaces in S _∞ ² ,the bounding ideal sphere of H ³,leading to a notion of constant mean curvature at infinity of H ³.A parametrization of all complete constant mean curvature surfaces at infinity of H ³ is given by holomorphic quadratic differentials on Ĉ,C, and D.     

Covering radii and paving diameters of Alexandrov spaces

Covering radii and paving diameters are defined, and the borderline case when cov_k X = π/2, k = 1,…,n + 1 and pav_k X = π/2, k = 1,…,n + 1 is studied (curv X ≥1, dim X = n).     

The initial value problem for cohomogeneity one Einstein metrics

The PDE Ric(g) = λ · g for a Riemannian Einstein metric g on a smooth manifold M becomes an ODE if we require g to be invariant under a Lie group G acting properly on M with principal orbits of codimension one. A singular orbit of the G-action gives a singularity of this ODE. Generically, an equation with such type of singularity has no smooth solution at the singularity. However, in our case, the very geometric nature of the equation makes it solvable. More precisely, we obtain a smooth G-invariant Einstein metric (with any Einstein constant λ) in a tubular neighbourhood around a singular orbit Q ⊂ M for any prescribed G-invariant metric g_Q and second fundamental form L_Q on Q, provided that the following technical condition is satisfied (which is very often the case): the representations of the principal isotropy group on the tangent and the normal space of the singular orbit Q have no common sub-representations. This Einstein metric is not uniquely determined by the initial data g_Q and L_Q; in fact, one may prescribe initial derivatives of higher degree, and examples show that this degree can be arbitrarily high. The proof involves a blend of ODE techniques and representation theory of the principal and singular isotropy groups.     

The one-phase Hele-Shaw problem with singularities

In this article we analyze viscosity solutions of the one phase Hele-Shaw problem in the plane and the corresponding free boundaries near a singularity. We find, up to order of magnitude, the speed at which the free boundary moves starting from a wedge, cusp, or finger-type singularity. Maximum principle-type arguments play a key role in the analysis.     

An exotic sphere with positive curvature almost everywhere

In this article we show that there is an exotic sphere with positive sectional curvature almost everywhere. In 1974 Gromoll and Meyer found a metric of nonnegative sectional on an exotic 7-sphere. They showed that the metric has positive curvature at a point and asserted, without proof, that the metric has positive sectional curvature almost everywhere [4]. We will show here that this assertion is wrong. In fact, the Gromoll-Meyer sphere has zero curvatures on an open set of points. Never the less, its metric can be perturbed to one that has positive curvature almost everywhere.     

On hyperbolically convex functions

Let be the unit disk of the complex plane. A conformai map of into itself is called hyperbolically convex if the non-Euclidean segment between any two points of also belongs to . In this paper we prove several inequalities that are analogous to inequalities about (Euclidean) convex univalent functions. We show that if ƒ (0) = 0, then Re zf′/f > 1/2. This inequality is the key for the results of this paper. In particular we deduce a three-variable inequality corresponding to that of Ruscheweyh and Sheil-Small. The sharp bound for the Schwarzian derivative remains open.     

Adapted complex structures in a neighborhood of an isotropic embedding

Let (X, ω) be a symplectic manifold and ι: M ? X an isotropic embedding, ι*ω = 0. The isotropie embedding theorem gives a local normal form of X in a neighborhood of M, in particular a natural potential α of ω, ?dα = ω. Now, given certain geometrical structures on M and on the symplectic normal bundle of M, in particular inducing a natural energy momentum function H in a neighborhood of M, we construct a natural complex structure J in a neighborhood of M satisfying certain initial conditions associated to the given initial data along M and satisfying the equation (in J): d^c H = α. This generalizes a theorem of Guillemin-Stenzel and Lempert-Szöke in the Lagrangean case.     

Proof of the double bubble curvature conjecture

An area minimizing double bubble in ℝⁿ is given by two (not necessarily connected) regions which have two prescribed n-dimensional volumes whose combined boundary has least (n−1)-dimensional area. The double bubble theorem states that such an area minimizer is necessarily given by a standard double bubble, composed of three spherical caps. This has now been proven for n = 2, 3,4, but is, for general volumes, unknown for n ≥ 5. Here, for arbitrary n, we prove a conjectured lower bound on the mean curvature of a standard double bubble. This provides an alternative line of reasoning for part of the proof of the double bubble theorem in ℝ³, as well as some new component bounds in ℝⁿ.     

The classification of complete orientable 2-dimensional area-minimizing mod 2 surfaces in ℝn

Let S ⊂ ℝⁿ be a complete 2-dimensional areaminimizing mod 2 surface. Then S = x₁ (M₁) ∪ … ∪ x_r (M_r) where each M_j is connected, x_j: M_j → V_j is a classical minimal immersion into an affine subspace V_j of ℝⁿ, and the subspaces V₁,…, V_r are pairwise orthogonal. Here we prove that if M_j is orientable, then x_j (M_j) is either aflat plane or, in suitable coordinates, a generalized complex hyperbola.     

Isoperimetric inequality in the Grushin plane

We prove a sharp isoperimetric inequality in the Grushin plane and compute the corresponding isoperimetric sets.     

Double bubbles in<Emphasis Type="Bold">S</Emphasis><Superscript>3</Superscript> and<Emphasis Type="Bold">H</Emphasis><Superscript>3</Superscript>

We prove the double bubble conjecture in the three-sphereS ³ and hyperbolic three-spaceH ³ in the cases where we can apply Hutchings theory:

Monotonicity inequalities for ther-area and a degeneracy theorem forr-minimal graphs

We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-minimal hypersurface in Euclidean space under appropriate quasi-positivity assumptions on certain invariants of the immersion. The proofs are based on the corresponding first variational formula. As an application, we derive a degeneracy theorem for an entire r-minimal graph whose defining function ƒ has first and second derivatives decaying fast enough at infinity: Its Hessian operator D² ƒ has at least n − r null eigenvalues everywhere.