Nonpositive Sectional Curvature Research Articles

The coarse geometric $\ell^p$-Novikov conjecture for subspaces of nonpositively curved manifolds

In this paper, we prove the coarse geometric \ell^p -Novikov conjecture for metric spaces with bounded geometry which admit a coarse embedding into a simply connected complete Riemannian manifold of nonpositive sectional curvature.

Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds with negative curvature

In this paper we obtain Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants on Riemannian manifolds with non-positive sectional curvature and, in particular, a variety of new estimates on hyperbolic spaces. Moreover, in some cases we also show their equivalence with Trudinger-Moser inequalities. As consequences, the relations between the constants of these inequalities are investigated yielding asymptotically best constants in the obtained inequalities. We also obtain the corresponding uncertainty type principles.

The space of almost calibrated (1,1)–forms on a compact Kähler manifold

The space $\mathcal{H}$ of "almost calibrated" $(1,1)$ forms on a compact K\"ahler manifold plays an important role in the study of the deformed Hermitian-Yang-Mills equation of mirror symmetry as emphasized by recent work of the second author and Yau, and is related by mirror symmetry to the space of positive Lagrangians studied by Solomon. This paper initiates the study of the geometry of $\mathcal{H}$. We show that $\mathcal{H}$ is an infinite dimensional Riemannian manifold with non-positive sectional curvature. In the hypercritical phase case we show that $\mathcal{H}$ has a well-defined metric structure, and that its completion is a ${\rm CAT}(0)$ geodesic metric space, and hence has an intrinsically defined ideal boundary. Finally, we show that in the hypercritical phase case $\mathcal{H}$ admits $C^{1,1}$ geodesics, improving a result of the second author and Yau. Using results of Darvas-Lempert we show that this result is sharp.

Path and quasi-homotopy for Sobolev maps between manifolds

We study the relationship between quasi-homotopy and path homotopy for Sobolev maps between manifolds. By employing singular integrals on manifolds we show that, in the critical exponent case, path homotopic maps are quasi-homotopic – and observe the rather surprising fact that quasi-homotopic maps need not be path homotopic. We also study the case where the target is an aspherical manifold, e.g. a manifold with non-positive sectional curvature, and the contrasting case of the target being a sphere.

Sharp Resolvent Estimates on Non-positively Curved Asymptotically Hyperbolic Manifolds

Abstract We study the high energy estimate for the resolvent $R(\lambda )$ of the Laplacian on non-trapping asymptotically hyperbolic manifolds (AHMs). In the literature, estimates of $R(\lambda )$ on weighted Sobolev spaces of the order $O(|\lambda |^{N})$ were established for some $N> -1$, $|\lambda |$ large, and $\lambda \in{{\mathbb{C}}}$ in strips where $R(\lambda )$ is holomorphic. In this work, we prove the optimal bound $O(|\lambda |^{-1})$ under the non-positive sectional curvature assumption by taking into account the oscillatory behavior of the Schwartz kernel of the resolvent.

Extremal Kähler Metrics Induced by Finite or Infinite-Dimensional Complex Space Forms

In this paper, we address the problem of studying those complex manifolds M equipped with extremal metrics g induced by finite or infinite-dimensional complex space forms. We prove that when g is assumed to be radial and the ambient space is finite-dimensional, then (M, g) is itself a complex space form. We extend this result to the infinite-dimensional setting by imposing the strongest assumption that the metric g has constant scalar curvature and is well behaved (see Definition 1 in the Introduction). Finally, we analyze the radial Kahler–Einstein metrics induced by infinite-dimensional elliptic complex space forms and we show that if such a metric is assumed to satisfy a stability condition then it is forced to have constant nonpositive holomorphic sectional curvature.

$L^{2p}_{f}$-harmonic $1$-forms on $f$-minimal hypersurfaces in a weighted manifold

In this paper, we prove the nonexistence of $L^{2p}_{f}$-harmonic $1$-forms on a complete noncompact $f$-minimal hypersurface in a weighted manifold with nonpositive sectional curvature. This results can be viewed as an extension of Yun and Seo's results on $L^{2}_{f}$-harmonic $1$-forms.

Holomorphic sectional curvature, nefness and Miyaoka–Yau type inequality

On a compact Kahler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kahler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kahler manifold of almost nonpositive holomorphic sectional curvature has a nef canonical line bundle, contains no rational curves and satisfies some Miyaoka-Yau type inequalities. In the course of the discussions, we attach a real value to any fixed Kahler class which, up to a constant factor depending only on the dimension of manifold, turns out to be an upper bound for the nef threshold.

Travelling times in scattering by obstacles in curved space

We consider travelling times of billiard trajectories in the exterior of an obstacle K on a two-dimensional Riemannian manifold M. We prove that given two obstacles with almost the same travelling times, the generalised geodesic flows on the non-trapping parts of their respective phase-spaces will have a time-preserving conjugacy. Moreover, if M has non-positive sectional curvature, we prove that if K and L are two obstacles with strictly convex boundaries and almost the same travelling times then K and L are identical.

Eells–Sampson Type Theorems for Subelliptic Harmonic Maps from sub-Riemannian Manifolds

In this paper, we consider critical maps of a horizontal energy functional for maps from a sub-Riemannian manifold to a Riemannian manifold. These critical maps are referred to as subelliptic harmonic maps. In terms of the subelliptic harmonic map heat flow, we investigate the existence problem for subelliptic harmonic maps. Under the assumption that the target Riemannian manifold has non-positive sectional curvature, we prove some Eells–Sampson type existence results for this flow when the source manifold is either a step-2 sub-Riemannian manifold or a step-r sub-Riemannian manifold whose sub-Riemannian structure comes from a tense Riemannian foliation. Finally, some Hartman type results are also established for the flow.

Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds

Let (M^n,g) be simply connected, complete, with non-positive sectional curvatures, and Sigma a 2-dimensional closed integral current (or flat chain mod 2) with compact support in M. Let S be an area minimising integral 3-current (resp. flat chain mod 2) such that partial S = Sigma . We use a weak mean curvature flow, obtained via elliptic regularisation, starting from Sigma , to show that S satisfies the optimal Euclidean isoperimetric inequality: 6 sqrt{pi }, mathbf {M}[S] le (mathbf {M}[Sigma ])^{3/2} . We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by -kappa < 0 and characterise the case of equality. The proof follows from an almost monotonicity of a suitable isoperimetric difference along the approximating flows in one dimension higher and an optimal estimate for the Willmore energy of a 2-dimensional integral varifold with first variation summable in L^2.

Plurisuperharmonicity of reciprocal energy function on Teichmüller space and Weil-Petersson metric

We consider harmonic maps u(z):Xz→N in a fixed homotopy class from Riemann surfaces Xz of genus g≥2 varying in the Teichmüller space T to a Riemannian manifold N with non-positive Hermitian sectional curvature. The energy function E(z)=E(u(z)) can be viewed as a function on T and we study its first and the second variations. We prove that the reciprocal energy function E(z)−1 is plurisuperharmonic on Teichmüller space. We also obtain the (strict) plurisubharmonicity of log⁡E(z) and E(z). As an application, we get the following relationship between the second variation of logarithmic energy function and the Weil-Petersson metric if the harmonic map u(z) is holomorphic or anti-holomorphic and totally geodesic, i.e.,(0.1)−1∂∂¯log⁡E(z)=ωWP2π(g−1). We consider also the energy function E(z) associated to the harmonic maps from a fixed compact Kähler manifold M to Riemann surfaces {Xz}z∈T in a fixed homotopy class. If u(z) is holomorphic or anti-holomorphic, then (0.1) is also proved.

Nonpositive curvature, the variance functional, and the Wasserstein barycenter

This paper connects nonpositive sectional curvature of a Riemannian manifold with the displacement convexity of the variance functional on the space $P(M)$ of probability measures over $M$. We show that $M$ has nonpositive sectional curvature and has trivial topology (i.e, is homeomorphic to $\mathbf{R}^n$) if and only if the variance functional on $P(M)$ is displacement convex. This is followed by a Jensen type inequality for the variance functional with respect to Wasserstein barycenters, as well as by a result comparing the variance of the Wasserstein and linear barycenters of a probability measure on $P(M)$ (that is, an element of $P(P(M))$). These results are applied to invariant measures under isometry group actions, giving a comparison for the variance functional between the Wasserstein projection and the $L^2$ projection to the set of invariant measures.

Rigidity of Busemann convex Finsler metrics

We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such metric is Riemannian or locally isometric to that of a normed plane. In the course of the proof we obtain new characterizations of Berwald metrics in terms of the so-called linear parallel transport.

Higher genera for proper actions of Lie groups

Let G be a Lie group with finitely many connected components and let K be a maximal compact subgroup. We assume that G satisfies the rapid decay (RD) property and that G/K has non-positive sectional curvature. As an example, we can take G to be a connected semisimple Lie group. Let M be a G-proper manifold with compact quotient M/G. In this paper we establish index formulae for the C^*-higher indices of a G-equivariant Dirac-type operator on M. We use these formulae to investigate geometric properties of suitably defined higher genera on M. In particular, we establish the G-homotopy invariance of the higher signatures of a G-proper manifold and the vanishing of the A-hat genera of a G-spin, G-proper manifold admitting a G-invariant metric of positive scalar curvature.

Chern-Ricci curvatures, holomorphic sectional curvature and Hermitian metrics

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal K\"{a}hler manifold with constant nonpositive holomorphic sectional curvature is K\"{a}hler. We also give examples of complete non-K\"{a}hler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature, and complete non-K\"{a}hler metric with zero holomorphic sectional curvature and nonvanishing curvature tensor.

RC-positivity and rigidity of harmonic maps into Riemannian manifolds

In this paper, we show that every harmonic map from a compact Kähler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, there is no non-constant harmonic map from a compact Kähler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.

Some regularity results for [formula omitted]-harmonic mappings between Riemannian manifolds

Let M be a C2-smooth Riemannian manifold with boundary and N a complete C2-smooth Riemannian manifold. We show that each stationary p-harmonic mapping u:M→N, whose image lies in a compact subset of N, is locally C1,α for some α∈(0,1), provided that N is simply connected and has non-positive sectional curvature. We also prove similar results for minimizing p-harmonic mappings with image being contained in a regular geodesic ball. Moreover, when M has non-negative Ricci curvature and N is simply connected with non-positive sectional curvature, we deduce a gradient estimate for C1-smooth weakly p-harmonic mappings from which follows a Liouville-type theorem in the same setting.

Convergence of subdivision schemes on Riemannian manifolds with nonpositive sectional curvature

This paper studies well-definedness and convergence of subdivision schemes which operate on Riemannian manifolds with nonpositive sectional curvature. These schemes are constructed from linear ones by replacing affine averages by the Riemannian centre of mass. In contrast to previous work, we consider schemes without any sign restriction on the mask, and our results apply to all input data. We also analyse the Hölder continuity of the resulting limit curves. Our main result states that if the norm of the derived scheme (resp. iterated derived scheme) is smaller than the corresponding dilation factor then the adapted scheme converges. In this way, we establish that convergence of a linear subdivision scheme is almost equivalent to convergence of its nonlinear manifold counterpart.

Explicit Bounds on Integrals of Eigenfunctions Over Curves in Surfaces of Nonpositive Curvature

Let (M, g) be a compact Riemannian surface with nonpositive sectional curvature and let $$\gamma $$ be a closed geodesic in M. And let $$e_\lambda $$ be an $$L^2$$-normalized eigenfunction of the Laplace–Beltrami operator $$\Delta _g$$ with $$-\Delta _g e_\lambda = \lambda ^2 e_\lambda $$. Sogge et al. (Camb J Math 5(1):123–151, 2017) showed using the Gauss–Bonnet Theorem that $$\int _\gamma e_\lambda \, \mathrm{{d}}s = O((\log \lambda )^{-1/2}),$$ an improvement over the general O(1) bound. We show this integral enjoys the same decay for a wide variety of curves, where M has nonpositive sectional curvature. These are the curves $$\gamma $$ whose geodesic curvature avoids, pointwise, the geodesic curvature of circles of infinite radius tangent to $$\gamma $$.