It is known that by dualizing the Bochner–Lichnerowicz–Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry–Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). When the manifold has a boundary, an appropriate generalization of the Reilly formula may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Brascamp–Lieb-type inequalities on the manifold. All previously known inequalities of Lichnerowicz, Brascamp–Lieb, Bobkov–Ledoux, and Veysseire are recovered, extended to the Riemannian setting and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are obtained. Our framework allows to encompass the entire class of Borell’s convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative generalized dimension.

We prove a new local inequality for divisors on surfaces and utilize it to compute α-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type A1 , A2 , A3 , A4 , A5 , or A6 are Kähler-Einstein.

We classify smooth del Pezzo surfaces whose α-invariant of Tian is bigger than 1.

We use the methods introduced by Cheltsov–Rubinstein–Zhang (Sel Math (N.S.) 25(2):25–34, 2019) to estimate δ-invariants of the seven singular del Pezzo surfaces with quotient singularities studied by Cheltsov–Park–Shramov (J Geom Anal 20(4):787–816, 2010) that have α-invariants less than 2/3. As a result, we verify that each of these surfaces admits an orbifold Kähler–Einstein metric.

Given a convex body K⊂RnK⊂Rn with the barycenter at the origin, we consider the corresponding Kähler–Einstein equation e−Φ=detD2Φe−Φ=detD2Φ. If *K* is a simplex, then the Ricci tensor of the Hessian metric D2ΦD2Φ is constant and equals n−14(n+1)n−14(n+1). We conjecture that the Ricci tensor of D2ΦD2Φfor an arbitrary convex body K⊆RnK⊆Rn is uniformly bounded from above by n−14(n+1)n−14(n+1) and we verify this conjecture in the two-dimensional case. The general case remains open.

Using Takahashi's theorem, we propose an approach to extend known families of minimal tori in spheres. As an example, the well-known two-parametric family of Lawson tau-surfaces including tori and Klein bottles is extended to a three-parametric family of tori and Klein bottles minimally immersed in spheres. Extremal spectral properties of the metrics on these surfaces are investigated. These metrics include (i) both metrics extremal for the first non-trivial eigenvalue on the torus, i.e., the metric on the Clifford torus and the metric on the equilateral torus and (ii) the metric maximal for the first non-trivial eigenvalue on the Klein bottle.

We consider an inverse problem for Laplacians on rotationally symmetric manifolds, which are finite for the transversal direction and periodic with respect to the axis of the manifold, i.e., Laplacians on tori. We construct an infinite dimensional analytic isomorphism between the space of profiles (the radius of the rotation) of the torus and the spectral data as well as the stability estimates: those for the spectral data in terms of the profile and conversely, for the profile in term of the spectral data.

The famous conjecture of Ivrii (Funct Anal Appl 14(2):98–106, 1980) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study its complex analytic version for quadrilateral orbits in two dimensions, with reflections from holomorphic curves. We present the complete classification of 4-reflective complex analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the author’s previous result Glutsyuk (Moscow Math J 14(2):239–289, 2014) classifying 4-reflective complex planar algebraic counterexamples. We provide applications to real planar billiards: classification of 4-reflective germs of real planar C4-smooth pseudo-billiards; solutions of Tabachnikov’s Commuting Billiard Conjecture and the 4-reflective case of Plakhov’s Invisibility Conjecture (both in two dimensions; the boundary is required to be piecewise C4-smooth).We provide a survey and a small technical result concerning higher number of complex reflections.

We study several of the recent conjectures in regards to the role of symmetry in the inequalities of Brunn-Minkowski type, such as the Lp-BrunnMinkowski conjecture of B¨or¨oczky, Lutwak, Yang and Zhang, and the Dimensional Brunn-Minkowski conjecture of Gardner and Zvavitch, in a unified framework. We obtain several new results for these conjectures. We show that when K ⊂ L, the multiplicative form of the Lp-Brunn-Minkowski conjecture holds for Lebesgue measure for p ≥ 1−Cn−0.75, which improves upon the estimate of Kolesnikov and Milman in the partial case when one body is contained in the other. We also show that the multiplicative version of the Lp-Brunn-Minkowski conjecture for the standard Gaussian measure holds in the case of sets containing sufficiently large ball (whose radius depends on p). In particular, the Gaussian Log-Brunn-Minkowski conjecture holds when K and L contain p 0.5(n + 1)Bn 2 . We formulate an a-priori stronger conjecture for log-concave measures, extending both the Lp-Brunn-Minkowski conjecture and the Dimensional one, and verify it in the case when the sets are dilates and the measure is Gaussian. We also show that the Log-Brunn-Minkowski conjecture, if verified, would yield this more general family of inequalities. Our results build up on the methods developed by Kolesnikov and Milman as well as Colesanti, Livshyts, Marsiglietti. We furthermore verify that the local version of these conjectures implies the global version in the setting of general measures, and this step uses methods developed recently by Putterman.

Let *M* be a complex manifold and *L* an oriented real line bundle on *M* equipped with a flat connection. A “locally conformally Kähler” (LCK) form is a closed, positive (1,1)-form taking values in *L*, and an LCK manifold is one which admits an LCK form. Locally, any LCK form is expressed as an *L*-valued pluri-Laplacian of a function called LCK potential. We consider a manifold *M* with an LCK form admitting an LCK potential (globally on *M*), and prove that *M *admits a positive LCK potential. Then *M* admits a holomorphic embedding to a Hopf manifold, as shown in Ornea and Verbitsky (Math Ann 348:25–33, 2010).

We provide a draft of a theory of geometric integration of “rough differential forms” which are generalizations of classical (smooth) differential forms to similar objects with very low regularity, for instance, involving Hölder continuous functions that may be nowhere differentiable. Borrowing ideas from the theory of rough paths, we show that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions. This can be seen as the basis of geometric integration similar to that used in geometric measure theory, but without any underlying differentiable structure, thus allowing Lipschitz functions and rectifiable sets to be substituted by far less regular objects (e.g. Hölder functions and their images which may be purely unrectifiable). Our construction includes both the one-dimensional Young integral and multidimensional integrals introduced recently by Züst, and provides also an alternative (and more geometric) view on the standard construction of rough paths. To simplify the exposition, we limit ourselves to integration of rough k-forms with k≤ 2.

We prove that (Formula presented.) and (Formula presented.) are the smallest log canonical thresholds of reduced plane curves of degree (Formula presented.), and we describe reduced plane curves of degree d whose log canonical thresholds are these numbers. As an application, we prove that (Formula presented.) and (Formula presented.) are the smallest values of the (Formula presented.)-invariant of Tian of smooth surfaces in (Formula presented.) of degree (Formula presented.). We also prove that every reduced plane curve of degree (Formula presented.) whose log canonical threshold is smaller than (Formula presented.) is GIT-unstable for the action of the group (Formula presented.), and we describe GIT-semistable reduced plane curves with log canonical thresholds (Formula presented.).