Volume 2024, Issue 810

We prove Bogomolov’s inequality on a normal projective variety in positive characteristic and we use it to show some new restriction theorems and a new boundedness result. We also redefine Higgs sheaves on normal varieties and we prove restriction theorems and Bogomolov-type inequalities for semistable logarithmic Higgs sheaves on some normal varieties in an arbitrary characteristic.

We introduce a new class of triangulated categories, which are Verdier quotients of three-Calabi–Yau categories from (decorated) marked surfaces, and show that its spaces of stability conditions can be identified with moduli spaces of framed quadratic differentials on Riemann surfaces with arbitrary order zeros and arbitrary higher order poles. A main tool in our proof is a comparison of two exchange graphs, obtained by tilting hearts in the quotient categories and by flipping mixed angulations associated with the quadratic differentials.

We prove that the linear syzygy spaces of a general canonical curve are spanned by syzygies of minimal rank.

This article studies typical 1-Lipschitz images of 𝑛-rectifiable metric spaces 𝐸 into R m \mathbb{R}^{m} for m ≥ n m\geq n . For example, if E ⊂ R k E\subset\mathbb{R}^{k} , we show that the Jacobian of such a typical 1-Lipschitz map equals 1 H n \mathcal{H}^{n} -almost everywhere and, if m > n m>n , preserves the Hausdorff measure of 𝐸. In general, we provide sufficient conditions, in terms of the tangent norms of 𝐸, for when a typical 1-Lipschitz map preserves the Hausdorff measure of 𝐸, up to some constant multiple. Almost optimal results for strongly 𝑛-rectifiable metric spaces are obtained. On the other hand, for any norm | ⋅ | \lvert\,{\cdot}\,\rvert on R m \mathbb{R}^{m} , we show that, in the space of 1-Lipschitz functions from ( [ − 1 , 1 ] n , | ⋅ | ∞ ) ([-1,1]^{n},\lvert\,{\cdot}\,\rvert_{\infty}) to ( R m , | ⋅ | ) (\mathbb{R}^{m},\lvert\,{\cdot}\,\rvert) , the H n \mathcal{H}^{n} -measure of a typical image is not bounded below by any Δ > 0 \Delta>0 .

In this article, we extend an unknottedness theorem for compact self-shrinkers to the mean curvature flow to shrinkers with finite topology and one asymptotically conical end, which conjecturally comprises the entire set of self-shrinkers with finite topology and one end. The mean curvature flow itself is used in the argument presented.

We prove that the Bergman kernel function associated to a smooth measure supported on a piecewise-smooth maximally totally real submanifold 𝐾 in C n \mathbb{C}^{n} is of polynomial growth. For example, this holds in dimension one if 𝐾 is a finite union of transverse Jordan arcs in ℂ. Our bounds are sharp when 𝐾 is smooth. We give an application to the equidistribution of the zeros of random polynomials, which extends a result of Shiffman–Zelditch to the higher-dimensional setting.

For a wide class of curvature energy functionals defined for planar curves under the fixed-length constraint, we obtain optimal necessary conditions for global and local minimizers. Our results extend Maddocks’ and Sachkov’s rigidity principles for Euler’s elastica by a new, unified and geometric approach. This in particular leads to complete classification of stable closed p -elasticae for all p ∈ ( 1 , ∞ ) {p\in(1,\infty)} and of stable pinned p -elasticae for p ∈ ( 1 , 2 ] {p\in(1,2]} . Our proof is based on a simple but robust “cut-and-paste” trick without computing the energy nor its second variation, which works well for planar periodic curves but also extends to some non-periodic or non-planar cases. An analytically remarkable point is that our method is directly valid for the highly singular regime p ∈ ( 1 , 3 2 ] {p\in(1,\frac{3}{2}]} in which the second variation may not exist even for smooth variations.

In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate: any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension.