Let $X$ be a nonsingular projective $n$-fold $(n \geq 2)$ which is either Fano or of general type with ample canonical bundle $K_X$ over an algebraic closed field $\kappa$ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes $c_1, c_2, \dotsc , c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of $\kappa$ is $0$, then the Chern ratios $\left( \dfrac{c_{2,1^{n-2}}}{c_{1^n}} , \dfrac{c_{2,2,1^{n-4}}}{c_{1^n}} , \dotsc , \frac{c_n}{c_{1^n}} \right)$ are contained in a convex polyhedron depending on the dimension of $X$ only. So we give an affirmative answer to a generalized open question, that whether the region described by the Chern ratios is bounded, posted by Hunt [Hun] to all dimensions. As a corollary, we can get that there exist constants $d_1$, $d_2$, $d_3$ and $d_4$ depending only on $n$ such that $d_1 K^n_X \leq \chi_\mathrm{top} (X) \leq d_2 K^n_X$ and $d_3 K^n_X \leq \chi (X, \mathcal{O}_X) \leq d_4 K^n_X$ . If the characteristic of $\kappa$ is positive, $K_X$ (or $-K_X$) is ample and $\mathcal{O}_X (K_X) (\mathcal{O}_X(-K_X) \textrm{, respectively})$ is globally generated, then the same results hold.

Let $X$ be a nonsingular projective $n$-fold $(n \geq 2)$ which is either Fano or of general type with ample canonical bundle $K_X$ over an algebraic closed field $\kappa$ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes $c_1, c_2, \dotsc , c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of $\kappa$ is $0$, then the Chern ratios $\left( \dfrac{c_{2,1^{n-2}}}{c_{1^n}} , \dfrac{c_{2,2,1^{n-4}}}{c_{1^n}} , \dotsc , \frac{c_n}{c_{1^n}} \right)$ are contained in a convex polyhedron depending on the dimension of $X$ only. So we give an affirmative answer to a generalized open question, that whether the region described by the Chern ratios is bounded, posted by Hunt [Hun] to all dimensions. As a corollary, we can get that there exist constants $d_1$, $d_2$, $d_3$ and $d_4$ depending only on $n$ such that $d_1 K^n_X \leq \chi_\mathrm{top} (X) \leq d_2 K^n_X$ and $d_3 K^n_X \leq \chi (X, \mathcal{O}_X) \leq d_4 K^n_X$ . If the characteristic of $\kappa$ is positive, $K_X$ (or $-K_X$) is ample and $\mathcal{O}_X (K_X) (\mathcal{O}_X(-K_X) \textrm{, respectively})$ is globally generated, then the same results hold.

Let us say that an $n$-sided polygon is semi-regular if it is circumscriptible, and its angles are all equal but possibly one which is then larger than the rest. Regular polygons, in particular, are semi-regular. The main result of the paper is that, in the class of convex polygons, semi-regular polygons are uniquely determined by just three geometric quantities: the area, the perimeter, and a third quantity depending only on the interior angles of the polygon that appears in the heat trace asymptotics of a polygon. As a consequence of this, we show that semi-regular polygons are spectrally determined, meaning that if $\Omega$ is a convex piecewise smooth planar domain, possibly with straight corners, whose Dirichlet or Neumann spectrum coincides with that of an $n$-sided semi-regular polygon $P$, then $\Omega$ is congruent to $P$.

Let us say that an $n$-sided polygon is semi-regular if it is circumscriptible, and its angles are all equal but possibly one which is then larger than the rest. Regular polygons, in particular, are semi-regular. The main result of the paper is that, in the class of convex polygons, semi-regular polygons are uniquely determined by just three geometric quantities: the area, the perimeter, and a third quantity depending only on the interior angles of the polygon that appears in the heat trace asymptotics of a polygon. As a consequence of this, we show that semi-regular polygons are spectrally determined, meaning that if $\Omega$ is a convex piecewise smooth planar domain, possibly with straight corners, whose Dirichlet or Neumann spectrum coincides with that of an $n$-sided semi-regular polygon $P$, then $\Omega$ is congruent to $P$.

We prove existence and uniqueness for a two-parameter family of translators for mean curvature flow. We get additional examples by taking limits at the boundary of the parameter space. Some of the translators resemble well-known minimal surfaces (Scherk’s doubly periodic minimal surfaces, helicoids), but others have no minimal surface analogs. A one-parameter subfamily of the examples (the pitchforks) have finite topology and quadratic area growth, and thus might arise as blowups at singularities of initially smooth, closed surfaces flowing by mean curvature flow.

We prove existence and uniqueness for a two-parameter family of translators for mean curvature flow. We get additional examples by taking limits at the boundary of the parameter space. Some of the translators resemble well-known minimal surfaces (Scherk’s doubly periodic minimal surfaces, helicoids), but others have no minimal surface analogs. A one-parameter subfamily of the examples (the pitchforks) have finite topology and quadratic area growth, and thus might arise as blowups at singularities of initially smooth, closed surfaces flowing by mean curvature flow.

The geometry of a ball within a Riemannian manifold is coarsely controlled if it has a lower bound on its Ricci curvature and a positive lower bound on its volume. We prove that such coarse local geometric control must persist for a definite amount of time under three-dimensional Ricci flow, and leads to local $C/t$ decay of the full curvature tensor, irrespective of what is happening beyond the local region.

As a by-product, our results generalise the Pseudolocality theorem of Perelman [19, §10.1 and §10.5] and Tian-Wang [25] in this dimension by not requiring the Ricci curvature to be almostpositive, and not asking the volume growth to be almost-Euclidean.

Our results also have applications to the topics of starting Ricci flow with manifolds of unbounded curvature, to the use of Ricci flow as a mollifier, and to the well-posedness of Ricci flow starting with Ricci limit spaces. In [24] we use results from this paper to prove that 3D Ricci limit spaces are locally bi-Hölder equivalent to smooth manifolds, going beyond a full resolution of the conjecture of Anderson, Cheeger, Colding and Tian in this dimension.

The geometry of a ball within a Riemannian manifold is coarsely controlled if it has a lower bound on its Ricci curvature and a positive lower bound on its volume. We prove that such coarse local geometric control must persist for a definite amount of time under three-dimensional Ricci flow, and leads to local $C/t$ decay of the full curvature tensor, irrespective of what is happening beyond the local region.

As a by-product, our results generalise the Pseudolocality theorem of Perelman [19, §10.1 and §10.5] and Tian-Wang [25] in this dimension by not requiring the Ricci curvature to be almostpositive, and not asking the volume growth to be almost-Euclidean.

Our results also have applications to the topics of starting Ricci flow with manifolds of unbounded curvature, to the use of Ricci flow as a mollifier, and to the well-posedness of Ricci flow starting with Ricci limit spaces. In [24] we use results from this paper to prove that 3D Ricci limit spaces are locally bi-Hölder equivalent to smooth manifolds, going beyond a full resolution of the conjecture of Anderson, Cheeger, Colding and Tian in this dimension.

In this paper, we show that for any closed $4$-dimensional simply-connected Riemannian manifold $M$ with Ricci curvature $\lvert \operatorname{Ric} \rvert \leq 3$, volume $\textrm{vol} (M) \gt v \gt 0$, and diameter $\textrm{diam} (M) \leq D$, the length of a shortest closed geodesic is bounded by a function $F(v,D)$ which only depends on $v$ and $D$.

The proofs of our result are based on a recent theorem of diffeomorphism finiteness of the manifolds satisfying the above conditions proven by J. Cheeger and A. Naber.

In this paper, we show that for any closed $4$-dimensional simply-connected Riemannian manifold $M$ with Ricci curvature $\lvert \operatorname{Ric} \rvert \leq 3$, volume $\textrm{vol} (M) \gt v \gt 0$, and diameter $\textrm{diam} (M) \leq D$, the length of a shortest closed geodesic is bounded by a function $F(v,D)$ which only depends on $v$ and $D$.

The proofs of our result are based on a recent theorem of diffeomorphism finiteness of the manifolds satisfying the above conditions proven by J. Cheeger and A. Naber.