We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section $\Sigma $ that we construct are either Birkhoff sections, which means that they intersect every sufficiently long orbit segment of the geodesic flow, or at least they have some hyperbolic components in $\partial \Sigma $ as limit sets of the orbits of the geodesic flow that do not return to $\Sigma $. In order to prove these theorems, we provide a study of configurations of simple closed geodesics of closed orientable Riemannian surfaces, which may have independent interest. Our arguments are based on the curve shortening flow.

Inspired by K. Fujita's algebro-geometric result that complex projective space has maximal degree among all K-semistable complex Fano varieties, we conjecture that the height of a K-semistable metrized arithmetic Fano variety $\mathcal {X}$ of relative dimension $n$ is maximal when $\mathcal {X}$ is the projective space over the integers, endowed with the Fubini–Study metric. Our main result establishes the conjecture for the canonical integral model of a toric Fano variety when $n\leq 6$ (the extension to higher dimensions is conditioned on a conjectural ‘gap hypothesis’ for the degree). Translated into toric Kähler geometry, this result yields a sharp lower bound on a toric invariant introduced by Donaldson, defined as the minimum of the toric Mabuchi functional. Furthermore, we reformulate our conjecture as an optimal lower bound on Odaka's modular height. In any dimension $n$ it is shown how to control the height of the canonical toric model $\mathcal {X},$ with respect to the Kähler–Einstein metric, by the degree of $\mathcal {X}$. In a sequel to this paper our height conjecture is established for any projective diagonal Fano hypersurface, by exploiting a more general logarithmic setup.

This article introduces the Clairaut conformal Riemannian map. This notion includes the previously studied notions of Clairaut conformal submersion, Clairaut Riemannian submersion, and the Clairaut Riemannian map as particular cases, and is well known in the classical theory of surfaces. Toward this, we find the necessary and sufficient condition for a conformal Riemannian map $\varphi : M \to N$ between Riemannian manifolds to be a Clairaut conformal Riemannian map with girth $s = e^f$. We show that the fibers of $\varphi $ are totally umbilical with mean curvature vector field the negative gradient of the logarithm of the girth function, that is, $-\nabla f$. Using this, we obtain a local splitting of M as a warped product and a usual product, if the horizontal space is integrable (under some appropriate hypothesis). We also provide some examples of the Clairaut conformal Riemannian maps to confirm our main theorem. We observe that the Laplacian of the logarithmic girth, that is, of f, on the total manifold takes the special form. It reduces to the Laplacian on the horizontal distribution, and if it is nonnegative, the universal covering space of M becomes a product manifold, under some hypothesis on f. Analysis of the Laplacian of f also yields the splitting of the universal covering space of M as a warped product under some appropriate conditions. We calculate the sectional curvature and mixed sectional curvature of M when f is a distance function. We also find the relationships between the total manifold and the fibers being symmetrical and, in particular, having constant sectional curvature, and from there, we compare their universal covering spaces, if fibers are also complete, provided f is a distance function. We also find a condition on the curvature tensor of the fibers to be semi-symmetric, provided that the total manifold is semi-symmetric and f is a distance function. In turn, this gives the warped product of symmetric, semi-symmetric spaces into two symmetric, semi-symmetric subspaces (under some hypothesis). Also if the Hessian or the Laplacian of the Riemannian curvature tensor fields is zero, or has a harmonic curvature tensor, then the fibers of $\varphi $ also satisfy the same property, if f is also a distance function. By obtaining Bochner-type formulas for Clairaut conformal Riemannian maps, we establish the relations between the divergences of the Ricci curvature tensor on fibers and horizontal space and the corresponding scalar curvature. We also study the horizontal Killing vector field of constant length and show that they are parallel under appropriate hypotheses. This in turn gives the splitting of the total manifold, if it admits a horizontal parallel Killing vector field and if the horizontal space is integrable. Finally, assuming that $\nabla f$ is a nontrivial gradient Ricci soliton on M, we prove that any vertical vector field is incompressible and hence the volume form of the fiber is invariant under the flow of the vector field.

For a proper, Gromov-hyperbolic metric space and a discrete, non-elementary, group of isometries, we define a natural subset of the limit set at infinity of the group called the ergodic limit set. The name is motivated by the fact that every ergodic measure which is invariant for the geodesic flow on the quotient metric space is concentrated on geodesics with endpoints belonging to the ergodic limit set. We refine the classical Bishop–Jones theorem proving that the packing dimension of the ergodic limit set coincides with the critical exponent of the group.

We classify hyperbolic polynomials in two real variables that admit a transitive action on some component of their hyperbolic level sets. Such surfaces are called special homogeneous surfaces, and they are equipped with a natural Riemannian metric obtained by restricting the negative Hessian of their defining polynomial. Independent of the degree of the polynomials, there exist a finite number of special homogeneous surfaces. They are either flat, or have constant negative curvature.

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of two-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle that covers the fusion of loops, i.e. the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish novel relations between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.

We show that any isometric immersion of a flat plane domain into ${\mathbb {R}}^3$ is developable provided it enjoys the little Hölder regularity $c^{1,2/3}$. In particular, isometric immersions of local $C^{1,\alpha }$ regularity with $\alpha >2/3$ belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss–Codazzi–Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge–Ampère equation analysed in [M. Lewicka and M. R. Pakzad. Anal. PDE 10 (2017), 695–727.].

Let G be a simply connected semisimple compact Lie group, let X be a simply connected compact Kähler manifold homogeneous under G, and let L be a negative holomorphic line bundle over X. We prove that all G-invariant Kähler metrics on the total space of L arise from the Calabi ansatz. Using this, we show that there exists a unique G-invariant scalar-flat Kähler metric in each G-invariant Kähler class of L. The G-invariant scalar-flat Kähler metrics are automatically asymptotically conical.

This paper is concerned with the study on an open problem of classifying conformally flat minimal Legendrian submanifolds in the $(2n+1)$-dimensional unit sphere $\mathbb {S}^{2n+1}$ admitting a Sasakian structure $(\varphi,\,\xi,\,\eta,\,g)$ for $n\ge 3$, motivated by the classification of minimal Legendrian submanifolds with constant sectional curvature. First of all, we completely classify such Legendrian submanifolds by assuming that the tensor $K:=-\varphi h$ is semi-parallel, which is introduced as a natural extension of $C$-parallel second fundamental form $h$. Secondly, such submanifolds have also been determined under the condition that the Ricci tensor is semi-parallel, generalizing the Einstein condition. Finally, as direct consequences, new characterizations of the Calabi torus are presented.

We give a sharp estimate for the first eigenvalue of the Schrödinger operator $L:=-\Delta -\sigma $ which is defined on the closed minimal submanifold $M^{n}$ in the unit sphere $\mathbb {S}^{n+m}$, where $\sigma $ is the square norm of the second fundamental form.

In Communications in Contemporary Mathematics 24 3, (2022),the authors have developed a method for constructing G-invariant partial differential equations (PDEs) imposed on hypersurfaces of an $(n+1)$-dimensional homogeneous space $G/H$, under mild assumptions on the Lie group G. In the present paper, the method is applied to the case when $G=\mathsf{PGL}(n+1)$ (respectively, $G=\mathsf{Aff}(n+1)$) and the homogeneous space $G/H$ is the $(n+1)$-dimensional projective $\mathbb{P}^{n+1}$ (respectively, affine $\mathbb{A}^{n+1}$) space, respectively. The main result of the paper is that projectively or affinely invariant PDEs with n independent and one unknown variables are in one-to-one correspondence with invariant hypersurfaces of the space of trace-free cubic forms in n variables with respect to the group $\mathsf{CO}(d,n-d)$ of conformal transformations of $\mathbb{R}^{d,n-d}$.

We show that for $n \neq 1,4$, the simplicial volume of an inward tame triangulable open $n$-manifold $M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also $\pi _1(M)$ is amenable, then the simplicial volume of $M$ vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.

The local structure of rotationally symmetric Finsler surfaces with vanishing flag curvature is completely determined in this paper. A geometric method for constructing such surfaces is introduced. The construction begins with a planar vector field X that depends on two functions of one variable. It is shown that the flow of X could be used to generate a generalized Finsler surface with zero flag curvature. Moreover, this generalized structure reduces to a regular Finsler metric if and only if X has an isochronous center. By relating X to a Liénard system, we obtain the isochronicity condition and discover numerous new examples of complete flat Finsler surfaces, depending on an odd function and an even function.

In this paper, we investigate the structure of certain solutions of the fully nonlinear Yamabe flow, which we call almost quotient Yamabe solitons as they extend quite naturally those already called quotient Yamabe solitons. We present sufficient conditions for a compact almost quotient Yamabe soliton to be either trivial or isometric with an Euclidean sphere. We also characterize noncompact almost gradient quotient Yamabe solitons satisfying certain conditions on both its Ricci tensor and potential function.

A version of the classical Buffon problem in the plane naturally extends to the setting of any Riemannian surface with constant Gaussian curvature. The Buffon probability determines a Buffon deficit. The relationship between Gaussian curvature and the Buffon deficit is similar to the relationship that the Bertrand–Diguet–Puiseux theorem establishes between Gaussian curvature and both circumference and area deficits.

We prove that there exists at least one positive Einstein metric on $\mathbb {HP}^{m+1}\sharp \overline {\mathbb {HP}}^{m+1}$ for $m\geq ~2$. Based on the existence of the first Einstein metric, we give a criterion to check the existence of a second Einstein metric on $\mathbb {HP}^{m+1}\sharp \overline {\mathbb {HP}}^{m+1}$. We also investigate the existence of cohomogeneity-one positive Einstein metrics on $\mathbb {S}^{4m+4}$ and prove the existence of a non-standard Einstein metric on $\mathbb {S}^8$.

We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to $L^p$-perturbations of their fundamental forms: For a manifold ${\mathcal M}^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immersions $f_n:{\mathcal M}^d\to {\mathcal N}^{d+1}$, whose pullback metrics and shape operators are arbitrary close in $L^p$ to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold ${\mathcal N}$, removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds.

We systematically study calibrated geometry in hyperkähler cones $C^{4n+4}$, their 3-Sasakian links $M^{4n+3}$, and the corresponding twistor spaces $Z^{4n+2}$, emphasizing the relationships between submanifold geometries in various spaces. Our analysis highlights the role played by a canonical $\mathrm {Sp}(n)\mathrm {U}(1)$-structure $\gamma $ on the twistor space Z. We observe that $\mathrm {Re}(e^{- i \theta } \gamma )$ is an $S^1$-family of semi-calibrations and make a detailed study of their associated calibrated geometries. As an application, we obtain new characterizations of complex Lagrangian and complex isotropic cones in hyperkähler cones, generalizing a result of Ejiri–Tsukada. We also generalize a theorem of Storm on submanifolds of twistor spaces that are Lagrangian with respect to both the Kähler–Einstein and nearly Kähler structures.

By using fixed point argument, we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g=\frac {da^2}{h(a^2)}+a^2g_{S^n}$ for some function h where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in $\mathbb {R}^n$ for any $n\ge 2$. More precisely, for any $\lambda \ge 0$ and $c_0>0$, we prove that there exist infinitely many solutions ${h\in C^2((0,\infty );\mathbb {R}^+)}$ for the equation $2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-\lambda r-(n-1))$, $h(r)>0$, in $(0,\infty )$ satisfying $\underset {\substack {r\to 0}}{\lim }\,r^{\sqrt {n}-1}h(r)=c_0$ and prove the higher-order asymptotic behavior of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behavior near the origin.

In this note, we establish a boundary maximum principle for a class of stationary pairs of varifolds satisfying a fixed contact angle condition in any compact Riemannian manifold with smooth boundary.