Generalizing previous constructions, we present a dual pair of decompositions of the complement of a link $L$ into bipyramids, given any multicrossing projection of $L$. When $L$ is hyperbolic, this gives new upper bounds on the volume of $L$ given its multicrossing projection. These bounds are realized by three closely related infinite tiling weaves.

We study non-flat planar 3‑webs with infinitesimal symmetries. Using multi-dimensional Schwarzian derivative we give a criterion for linearization of such webs and present a projective classification thereof. Using this classification we show that the Gronwall conjecture is true for 3‑webs admitting infinitesimal symmetries.

The purpose of this article is to investigate the structure of complete non-compact quasi-Einstein manifolds. We show that complete noncompact quasi-Einstein manifolds with $\lambda = 0$ are connected at infinity. In addition, we provide some conditions under which quasi-Einstein manifolds with $\lambda \lt 0$ are $f$-non-parabolic. In particular, we obtain estimates on volume growth of geodesic balls

In this paper, we generalize Cao–Yau’s gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li–Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As

We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by ambient Ricci curvature or, in the non-Kähler case, by its analogues. To this end we explore the geometry of totally real submanifolds, defining (i) a new geometric

By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional $(d \geq 3)$ Riemannian manifold without boundary. As a direct consequence of our main theorems, we prove the existence of at least one solution to the following Yamabe-type problem\[\begin{cases}-\Delta_g w + \alpha(\sigma) w = \mu K(\sigma) w^{\frac{d+2}{d-2}}

We consider solutions $(M^4 , g(t)), 0 \leq t \lt T$, to Ricci flow on compact, four-dimensional manifolds without boundary. We prove integral curvature estimates which are valid for any such solution. In the case that the scalar curvature is bounded and $T \lt \infty$, we show that these estimates imply that the (spatial) integral of the square of the norm of the Riemannian curvature is bounded by

In this paper, we will prove a gap theorem on four-dimensional gradient shrinking soliton. More precisely, we will show that any complete four-dimensional gradient shrinking soliton with nonnegative and bounded Ricci curvature, satisfying a pinched Weyl curvature, either is flat, or $\lambda_1 + \lambda_2 \geq c_0 R \gt 0$ at all points, where $c_0 \approx 0.29167$ and $\lbrace \lambda_i \rbrace$ are