Volume 25, Issue 1

We prove a new rigidity result for metrics defined on closed smooth n -manifolds that are critical for the quadratic functional 𝔉 t , which depends on the Ricci curvature Ric and the scalar curvature R , and that satisfy a pinching condition of the form Sec > ε R , where ε is a function of t and n , while Sec denotes the sectional curvature. In particular, we show that Bach-flat metrics with constant scalar curvature satisfying Sec > 148 $\begin{array}{} \displaystyle \frac{1}{48} \end{array}$ R are Einstein and, by a known result, are isometric to 𝕊 4 , ℝℙ 4 or ℂℙ 2 .

Mixed volumes in n -dimensional Euclidean space are functionals of n -tuples consisting of convex bodies K , L , C 1 , …, C n –2 . The Alexandrov–Fenchel inequalities are fundamental inequalities between mixed volumes of convex bodies, which cover as very special cases many important inequalities between basic geometric functionals. The problem of characterizing completely the equality cases in the Alexandrov–Fenchel inequality is wide open. Major recent progress was made by Yair Shenfeld and Ramon van Handel [25; 26], in particular they resolved the problem in the cases where K , L are general convex bodies and C 1 , …, C n –2 are polytopes, zonoids or smooth bodies (under some dimensional restriction). We introduce the class of polyoids, which includes polytopes, zonoids and triangle bodies, and characterize polyoids by using generating measures. Based on this characterization and Shenfeld and van Handel's contribution, we extend their result to a class of convex bodies containing all polyoids and smooth bodies (under some dimensional restriction). Our result is stated in terms of the support of the mixed area measure of the unit ball B n and convex bodies C 1 , …, C n –2 . A geometric description of this support is provided in the accompanying work [12].

In this paper, we first derive the covariant tensorial formula for the asymptotic operators of contact instantons w : Σ̇ → Q and of pseudoholomorphic curves u : ( Σ̇ , j ) → ( Q × ℝ, J͠ ) on the symplectization of a contact manifold ( Q , λ ). The formula is independent of the choice of connection on the nose and exhibits explicit dependence on the compatible pair ( λ , J ) of a given contact triad ( Q , λ , J ). Then based on this, we prove that all eigenvalues of the asymptotic operator are simple for a generic choice of compatible CR almost complex structure J for a given contact form λ . This perturbation theory has been missing in the study of pseudoholomorphic curves on symplectization.

We find a criterion that completely characterizes the existence of ample polar cylinders of Fujita rank 2 in smooth del Pezzo surfaces of degree 2.

In this paper, we give a generalization of Fenchel’s theorem for closed curves as frontals in Euclidean space ℝ n . We prove that, for a non-co-orientable closed frontal in ℝ n , its total absolute curvature is greater than or equal to π . It is equal to π if and only if the curve is a planar locally L -convex closed frontal whose rotation index is 1/2 or –1/2. Furthermore, if the equality holds and if every singular point is a cusp, then the number N of cusps is an odd integer greater than or equal to 3, and N = 3 holds if and only if the curve is simple.

We establish some relations between the perimeter, the area and the visual angle of a planar compact convex set. Our first result states that Crofton’s formula is the unique universal formula relating the visual angle, the length and the area. After that we give a characterization of convex sets of constant width by means of the behavior of their isotopic sets at infinity. Also for this class of convex sets we prove that the existence of an isotopic circle is enough to ensure that the considered set is a disc.

For 𝓞 an imaginary quadratic ring, we compute a fundamental polyhedron in hyperbolic 3-space for the action of PE 2 (𝓞), the projective elementary subgroup of PSL 2 (𝓞). This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for PE 2 (𝓞), show that it has infinite index and is its own normalizer in PSL 2 (𝓞), and split PSL 2 (𝓞) into a free product with amalgamation that has PE 2 (𝓞) as one of its factors.

In this article, we combine sums of squares (SOS) and sums of nonnegative circuit (SONC) forms, two independent nonnegativity certificates for real homogeneous polynomials. We consider the convex cone SOS+SONC of forms that decompose into a sum of an SOS and a SONC form and study it from a geometric point of view. We show that the SOS+SONC cone is proper and neither closed under multiplication nor under linear transformation of variables. Moreover, we present an alternative proof of an analog of Hilbert’s 1888 Theorem for the SOS+SONC cone and prove that in the non-Hilbert cases it provides a proper superset of the union of the SOS and SONC cones. This follows by exploiting a new necessary condition for membership in the SONC cone.