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In this paper, we study a Serrin-type partially overdetermined problem proposed by Guo-Xia [Calc. Var. Partial Differential Equations 58 (2019), Paper No. 160, 15], and prove a rigidity result that characterizes capillary spherical caps in a half ball. [ABSTRACT FROM AUTHOR]

We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwines with a new method for computing an explicit basis of the singular homology of the hypersurface. It is based on Picard–Lefschetz theory and relies on the computation of the monodromy action induced by a one-parameter family of hyperplane sections on the homology of a given section. We provide a SageMath implementation. For example, on a laptop, it makes it possible to compute the periods of a smooth complex quartic surface with hundreds of digits of precision in typically an hour. [ABSTRACT FROM AUTHOR]

In this note, Gromov's reduction [ No metrics with positive scalar curvatures on aspherical 5-manifolds , https://arxiv.org/abs/2009.05332, 2020], from the aspherical conjecture to the generalized filling radius conjecture, is generalized to the smooth \mathbb Q-homology vanishing conjecture in the case of hypersurface. In particular, we can show that any continuous map from a closed 4-manifold admitting positive scalar curvature to an aspherical 5-manifold induces zero map between H_4(\cdot,\mathbb Q). As a corollary, we obtain the following aspherical splitting theorem: if a complete orientable aspherical Riemannian 5-manifold has non-negative scalar curvature and two ends, then it splits into the Riemannian product of a closed flat manifold and the real line. [ABSTRACT FROM AUTHOR]

This paper establishes a necessary and sufficient condition for L^p-boundedness of a class of multilinear functionals which includes both the Brascamp-Lieb inequalities and generalized Radon transforms associated to algebraic incidence relations. The testing condition involves bounding the average of an inverse power of certain Jacobian-type quantities along fibers of associated projections and covers many widely-studied special cases, including convolution with measures on nondegenerate hypersurfaces or on nondegenerate curves. The heart of the proof is based on Guth's visibility lemma [Acta Math. 205 (2010), pp. 263–286] in one direction and on a careful analysis of Knapp-type examples in the other. Various applications are discussed which demonstrate new and subtle interplay between curvature and transversality and establish nontrivial mixed-norm L^p-improving inequalities in the model case of convolution with affine hypersurface measure on the paraboloid. [ABSTRACT FROM AUTHOR]

Let F \in \mathbb {Z}[x_0, \ldots, x_n] be homogeneous of degree d and assume that F is not a 'nullform', i.e., there is an invariant I of forms of degree d in n+1 variables such that I(F) \neq 0. Equivalently, F is semistable in the sense of Geometric Invariant Theory. Minimizing F at a prime p means to produce T \in Mat(n+1, \mathbb {Z}) \cap GL(n+1, \mathbb {Q}) and e \in \mathbb {Z}_{\ge 0} such that F_1 = p^{-e} F([x_0, \ldots, x_n] \cdot T) has integral coefficients and v_p(I(F_1)) is minimal among all such F_1. Following Kollár [Electron. Res. Announc. Amer. Math. Soc. 3 (1997), pp. 17–27], the minimization process can be described in terms of applying weight vectors w \in \mathbb {Z}_{\ge 0}^{n+1} to F. We show that for any dimension n and degree d, there is a complete set of weight vectors consisting of [0,w_1,w_2,\dots,w_n] with 0 \le w_1 \le w_2 \le \dots \le w_n \le 2 n d^{n-1}. When n = 2, we improve the bound to d. This answers a question raised by Kollár. These results are valid in a more general context, replacing \mathbb {Z} and p by a PID R and a prime element of R. Based on this result and a further study of the minimization process in the planar case n = 2, we devise an efficient minimization algorithm for ternary forms (equivalently, plane curves) of arbitrary degree d. We also describe a similar algorithm that allows to minimize (and reduce) cubic surfaces. These algorithms are available in the computer algebra system Magma. [ABSTRACT FROM AUTHOR]

In this note, we prove a family of sharp weighed inequalities which involve weighted k-th mean curvature integral and two distinct quermassintegrals for closed hypersurfaces in \mathbb {R}^n. This inequality generalizes the corresponding result of Wei and Zhou [Bull. Lond. Math. Soc. 55 (2023), pp. 263–281] where their proof is based on earlier results of Kwong-Miao [Pacific J. Math. 267 (2014), pp. 417–422; Commun. Contemp. Math. 17 (2015), p. 1550014]. Here we present a proof which does not rely on Kwong-Miao's results. [ABSTRACT FROM AUTHOR]

Classifying the nonflat hypersurfaces in Euclidean space f\colon M^n\to \mathbb {R}^{n+1} that locally admit smooth infinitesimal deformations that preserve the Gauss map infinitesimally was a problem only considered by Schouten in 1928 [Proceedings Amsterdam 31 (1928), pp. 208–218]. He found two conditions that are necessary and sufficient, with the first one being the minimality of the submanifold. The second is a technical condition that does not clarify much about the geometric nature of the hypersurface. In that respect, the parametric solution of the problem given in this note yields that the submanifold has to be Kaehler. [ABSTRACT FROM AUTHOR]

Using a refinement of the differential method introduced by Oguiso and Yu, we provide effective conditions under which the automorphisms of a smooth degree d hypersurface of \mathbf {P}^{n+1} are given by generalized triangular matrices. Applying this criterion we compute all the remaining automorphism groups of Klein hypersurfaces of dimension n\geq 1 and degree d\geq 3 with (n,d)\neq (2,4). We introduce the concept of extremal polarized Hodge structures, which are structures that admit an automorphism of large prime order. Using this notion, we compute the automorphism group of the polarized Hodge structure of certain Klein hypersurfaces that we call of Wagstaff type, which are characterized by the existence of an automorphism of large prime order. For cubic hypersurfaces and some other values of (n,d), we show that both groups coincide (up to involution) as predicted by the Torelli Principle. [ABSTRACT FROM AUTHOR]

For any Q\in \{\frac {3}{2},2,\frac {5}{2},3,\dotsc \}, we establish a structure theory for the class \mathcal {S}_Q of stable codimension 1 stationary integral varifolds admitting no classical singularities of density

We study local-global principles for two notions of semi-integral points, termed Campana points and Darmon points. In particular, we develop a semi-integral version of the Brauer–Manin obstruction interpolating between Manin's classical version for rational points and the integral version developed by Colliot-Thélène and Xu. We determine the status of local-global principles, and obstructions to them, in two families of orbifolds naturally associated to quadric hypersurfaces. Further, we establish a quantitative result measuring the failure of the semi-integral Brauer–Manin obstruction to account for its integral counterpart for affine quadrics. [ABSTRACT FROM AUTHOR]

We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space \mathbb {H}^{n+1} (n\geq 2) with the speed given by arbitrary positive power \alpha of the Gauss curvature. We prove that if the initial hypersurface is convex, then the smooth solution of the flow remains convex and exists for all positive time t\in [0,\infty). Moreover, we apply a result of Kohlmann which characterises the geodesic ball using the hyperbolic curvature measures and an argument of Alexandrov reflection to prove that the flow converges to a geodesic sphere exponentially in the smooth topology. This can be viewed as the first result for non-local type volume preserving curvature flows for hypersurfaces in the hyperbolic space with only convexity required on the initial data. [ABSTRACT FROM AUTHOR]

In this short note we see that double-well phase transitions exhibit more rigidity than their minimal hypersurface counterparts. [ABSTRACT FROM AUTHOR]

In this paper, we study 4-dimensional complete hypersurfaces with w-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for 4-dimensional complete hypersurfaces with w-constant mean curvature. As a by-product, we give a new proof of the result of Deng-Gu-Wei [Adv. Math. 314 (2017), pp. 278–305] under the weaker topological condition. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions to the centroaffine Minkowski problem, namely the existence of closed convex hypersurfaces in the Euclidean space \mathbb {R}^{n+1} with prescribed centroaffine curvature. This problem can be described as a variational problem with a functional resembling Kantorovich's dual functional in optimal transportation. By using the Gauss curvature flow, we obtain new conditions for the existence of solutions to the problem. [ABSTRACT FROM AUTHOR]

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties. [ABSTRACT FROM AUTHOR]

In this paper, we solve the asymptotic Plateau problem in hyperbolic space for constant \sigma _{n-1} curvature, i.e. the existence of a complete hypersurface in \mathbb {H}^{n+1} satisfying \sigma _{n-1}(\kappa)=\sigma \in (0,n) with a prescribed asymptotic boundary \Gamma. The key ingredient is the curvature estimates. Previously, this was only known for \sigma _0<\sigma

We apply the Murnaghan-Nakayama rule in the representation theory of symmetric groups to develop new techniques for studying biharmonic hypersurfaces in a space form. As applications of the new techniques, we settle the well-known Chen's conjecture on biharmonic hypersurfaces in \Bbb R^6 and BMO conjecture on biharmonic hypersurfaces in \mathbb S^6. [ABSTRACT FROM AUTHOR]

We introduce a new generalization of the notion of preperiodic hypersurface and explore some of its basic ramifications. We also prove that among nonlinear endomorphisms of projective space, those with a periodic critical point are Zariski dense. This answers a question of Ingram. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the first and second widths of the real projective space \mathbb {RP}^{n}, for n ranging from 4 to 7, and for this we used some tools from the Almgren-Pitts min-max theory. In a recent paper, Ramirez-Luna computed the first width of the real projective spaces, and, at the same time, we obtained optimal sweepouts realizing the first and second widths of those spaces. [ABSTRACT FROM AUTHOR]

The classical classifications of the locally isometrically deformable Euclidean hypersurfaces obtained by U. Sbrana in 1909 and E. Cartan in 1916 include four classes, among them the one formed by submanifolds that allow just a single deformation. The question of whether these Sbrana-Cartan hypersurfaces do, in fact, exist was not addressed by either of them. Positive answers to this question were given by Dajczer-Florit-Tojeiro in 1998 for the ones called of hyperbolic type and by Dajczer-Florit in 2004 when of elliptic type which is the other possibility. In both cases the examples constructed are rather special. The main result of this paper yields an abundance of examples of hypersurfaces of either type and seems to point in the direction of a classification although that goal remains elusive. [ABSTRACT FROM AUTHOR]

Let \mathbb {F}_q be a finite field with q=p^n elements. In this paper, we study the number of \mathbb {F}_q-rational points on the affine hypersurface \mathcal X given by a_1 x_1^{d_1}+\dots +a_s x_s^{d_s}=b, where b\in \mathbb {F}_q^*. A classic well-known result of Weil yields a bound for such number of points. This paper presents necessary and sufficient conditions for the maximality and minimality of \mathcal X with respect to Weil's bound. [ABSTRACT FROM AUTHOR]

In this paper, we study a locally constrained mean curvature flow with free boundary in a hyperbolic ball. Under the flow, the enclosed volume is preserved and the area is decreasing. We prove the long time existence and smooth convergence for such flow under certain star-shaped condition. As an application, we give a flow proof of the isoperimetric problem for the star-shaped free boundary hypersurfaces in a hyperbolic ball. [ABSTRACT FROM AUTHOR]

In this paper, we prove a class of weighted isoperimetric inequalities for bounded domains in hyperbolic space by using the isoperimetric inequality with log-convex density in Euclidean space. As a consequence, we remove the horo-convex assumption of domains in a weighted isoperimetric inequality proved by Scheuer-Xia [Trans. Amer. Math. Soc. 372 (2019), pp. 6771–6803]. Furthermore, we prove weighted isoperimetric inequalities for star-shaped domains in warped product manifolds. Particularly, we obtain a weighted isoperimetric inequality for star-shaped hypersurfaces lying outside a certain radial coordinate slice in the anti-de Sitter-Schwarzschild manifold. [ABSTRACT FROM AUTHOR]

Let M^n be a closed hypersurface with constant mean curvature and constant scalar curvature in the unit sphere \mathbb {S}^{n+1}. Denote by S and H the squared length of the second fundamental form and the mean curvature of M, respectively. For a fixed integer k, where 1\leq k\leq n-2, we prove that there exists a positive constant \gamma (n) depending only on n, such that if |H|<\gamma (n) and \alpha _k(n,H)\leq S \leq \alpha _{k+1}(n,H), then either S=\alpha _k(n,H) and M must be the Clifford torus \mathbb {S}^{n-k}\big (\frac {1}{\sqrt {1+\lambda _k^2}}\big)\times \mathbb {S}^{k}\big (\frac {\lambda _k}{\sqrt {1+\lambda _k^2}}\big), or S=\alpha _{k+1}(n,H) and M is the Clifford torus \mathbb {S}^{n-k-1}\big (\frac {1}{\sqrt {1+\lambda _{k+1}^2}}\big)\times \mathbb {S}^{k+1}\big (\frac {\lambda _{k+1}}{\sqrt {1+\lambda _{k+1}^2}}\big). Here \alpha _k(n,H)=n+\frac {n^3}{2k(n-k)}H^2-\frac {n(n-2k)}{2k(n-k)}\sqrt {n^2H^4+4k(n-k)H^2} and \lambda _k=\frac {n|H|+\sqrt {n^2H^2+4k(n-k)}}{2(n-k)}. [ABSTRACT FROM AUTHOR]

A real hypersurface in \mathbb {C}^2 is said to be Reinhardt if it is invariant under the standard \mathbb {T}^2-action on \mathbb {C}^2. Its CR geometry can be described in terms of the curvature function of its "generating curve", i.e., the logarithmic image of the hypersurface in the plane \mathbb {R}^2. We give a sharp upper bound for the first positive eigenvalue of the Kohn Laplacian associated to a natural pseudohermitian structure on a compact and strictly pseudoconvex Reinhardt real hypersurface having closed generating curve (which amounts to the \mathbb {T}^2-action being free). Our bound is expressed in terms of the L^2-norm of the curvature function of the generating curve and is attained if and only if the curve is a circle. [ABSTRACT FROM AUTHOR]

In this paper, we consider a large class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space \mathbb {R}^{n+1} with speed \psi u^\alpha \rho ^\delta f^{-\beta }, where \psi is a smooth positive function on unit sphere, u is the support function of the hypersurface, \rho is the radial function, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. When \psi =1, we prove that the flow exists for all time and converges to infinity if \alpha +\delta +\beta \leqslant 1, and \alpha \leqslant 0<\beta, while in case \alpha +\delta +\beta >1, \alpha,\delta \leqslant 0<\beta, the flow blows up in finite time, and where we assume the initial hypersurface to be strictly convex. In both cases the properly rescaled flows converge to a sphere centered at the origin. In particular, the results of Gerhardt [J. Differential Geom. 32 (1990), pp. 299–314; Calc. Var. Partial Differential Equations 49 (2014), pp. 471–489] and Urbas [Math. Z. 205 (1990), pp. 355–372] can be recovered by putting \alpha =\delta =0. Our previous works [Proc. Amer. Math. Soc. 148 (2020), pp. 5331–5341; J. Funct. Anal. 282 (2022), p. 38] and Hu, Mao, Tu and Wu [J. Korean Math. Soc. 57 (2020), pp. 1299–1322] can be recovered by putting \delta =0 and \alpha =0 respectively. By the convergence of these flows, we can give a new proof of uniqueness theorems for solutions to L^p-Minkowski problem and L^p-Christoffel-Minkowski problem with constant prescribed data. Similarly, we consider the L^p dual Christoffel-Minkowski problem and prove a uniqueness theorem for solutions to L^p dual Minkowski problem and L^p dual Christoffel-Minkowski problem with constant prescribed data. At last, we focus on the long time existence and convergence of a class of anisotropic flows (i.e. for general function \psi). The final result not only gives a new proof of many previously known solutions to L^p dual Minkowski problem, L^p-Christoffel-Minkowski problem, etc. by such anisotropic flows, but also provides solutions to L^p dual Christoffel-Minkowski problem with some conditions. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the quermassintegrals for convex hypersurfaces with capillary boundary in the unit Euclidean ball {\mathbb {B}}^{n+1} and derive its first variational formula. Then by using a locally constrained nonlinear curvature flow, which preserves the n-th quermassintegral and non-decreases the k-th quermassintegral, we obtain the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in {\mathbb {B}}^{n+1}. This generalizes the result of Scheuer [J. Differential Geom. 120 (2022), pp. 345–373] for convex hypersurfaces with free boundary in {\mathbb {B}}^{n+1}. [ABSTRACT FROM AUTHOR]

We develop a version of Mikhalkin's lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a \delta dimensional linear space of degree d real hypersurfaces containing \frac {1}{\delta !}(\gamma _nd^n)^{\delta }+O(d^{n\delta -1}) hypersurfaces with \delta real nodes, where \gamma _n are positive and given by a recursive formula. This is asymptotically comparable to the number \frac {1}{\delta !} \left ((n+1)(d-1)^n \right)^{\delta }+O\left (d^{n(\delta -1)} \right) of complex hypersurfaces having \delta nodes in a \delta dimensional linear space. In the case \delta =1 we give a slightly better leading term. [ABSTRACT FROM AUTHOR]

We consider the tropicalization of tangent lines to a complete intersection curve X in \mathbb {P}^n. Under mild hypotheses, we describe a procedure for computing the tropicalization of the image of the Gauss map of X in terms of the tropicalizations of the hypersurfaces cutting out X. We apply this to obtain descriptions of the tropicalization of the dual variety X^* and tangential variety \tau (X) of X. In particular, we are able to compute the degrees of X^* and \tau (X) and the Newton polytope of \tau (X) without using any elimination theory. [ABSTRACT FROM AUTHOR]

In this paper, regarding the Riemannian product \mathbb {S}^2\times \mathbb {S}^2 of two unit 2-spheres as a Kähler surface, we study its real hypersurfaces with typical geometric properties. First, we classify the real hypersurfaces of \mathbb {S}^2\times \mathbb {S}^2 with isometric Reeb flow and then, by using a Simons' type inequality, a characterization of these compact real hypersurfaces is provided. Next, we classify Hopf hypersurfaces of \mathbb {S}^2\times \mathbb {S}^2 with constant product angle function. Finally, we classify Hopf hypersurfaces of \mathbb {S}^2\times \mathbb {S}^2 with parallel Ricci tensor. [ABSTRACT FROM AUTHOR]

Let \mathcal {A} denote a central hyperplane arrangement of rank n in affine space \mathbb {K}^n over a field \mathbb {K} of characteristic zero and let l_1,\ldots, l_m\in R≔\mathbb {K}[x_1,\ldots,x_n] denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial f≔l_1\cdots l_m\in R. The focus of the paper is on the ideal J_f\subset R generated by the partial derivatives of f. We conjecture that J_f is a minimal reduction of the ideal \mathbb {I}\subset R generated by the (m-1)-fold products of distinct forms among l_1,\ldots, l_m. We prove this conjecture for an almost generic \mathcal {A} (i.e., any n-1 among the defining linear forms are linearly independent). In this case we obtain a stronger version of a result by Dimca and Papadima, and we confirm the conjecture unconditionally for n=3. We also conjecture that J_f is an ideal of linear type (i.e., the respective symmetric and Rees algebras coincide). We prove this conjecture for n=3. In the sequel we explain the tight relationship between the two ideals J_f, \mathbb {I}\subset R; in particular, we show that in the generic case (J_f)^{\text {sat}}=\mathbb I. As a consequence, we can provide a simpler proof of a conjectured result of Yuzvinsky, proved by Rose and Terao, on the vanishing of the depth of R/J_f. [ABSTRACT FROM AUTHOR]

We provide a formula for F-thresholds of a Thom-Sebastiani type polynomial over a perfect field of prime characteristic. We also compute the first test ideal of Thom-Sebastiani type polynomials. Finally, we apply our results to find hypersurfaces where the log canonical thresholds equal the F-pure thresholds for infinitely many prime numbers. [ABSTRACT FROM AUTHOR]

We introduce and prove the n-dimensional Pizza Theorem: Let \mathcal {H} be a hyperplane arrangement in \mathbb {R}^{n}. If K is a measurable set of finite volume, the pizza quantity of K is the alternating sum of the volumes of the regions obtained by intersecting K with the arrangement \mathcal {H}. We prove that if \mathcal {H} is a Coxeter arrangement different from A_{1}^{n} such that the group of isometries W generated by the reflections in the hyperplanes of \mathcal {H} contains the map -\mathrm {id}, and if K is a translate of a convex body that is stable under W and contains the origin, then the pizza quantity of K is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of \mathcal {H} that we call the even restricted arrangement. More generally, we prove that for a class of arrangements that we call even (this includes the Coxeter arrangements above) and for a sufficiently symmetric set K, the pizza quantity of K+a is polynomial in a for a small enough, for example if K is convex and 0\in K+a. We get stronger results in the case of balls, more generally, convex bodies bounded by quadratic hypersurfaces. For example, we prove that the pizza quantity of the ball centered at a having radius R\geq \|a\| vanishes for a Coxeter arrangement \mathcal {H} with |\mathcal {H}|-n an even positive integer. We also prove the Pizza Theorem for the surface volume: When \mathcal {H} is a Coxeter arrangement and |\mathcal {H}| - n is a nonnegative even integer, for an n-dimensional ball the alternating sum of the (n-1)-dimensional surface volumes of the regions is equal to zero. [ABSTRACT FROM AUTHOR]

We prove a weak version of the Perdomo Conjecture, namely, there is a positive constant \delta (n)>0 depending only on n such that on any closed embedded, non-totally geodesic, minimal hypersurface M^n in \mathbb {S}^{n+1}, \begin{equation*} \int _{M}S \geq \delta (n)\operatorname {Vol}(M^n), \end{equation*} where S is the squared length of the second fundamental form of M^n. The Perdomo Conjecture asserts that \delta (n)=n which is still open in general. As byproducts, we also obtain some integral inequalities and Simons-type pinching results on closed embedded (or immersed) minimal hypersurfaces, with the first positive eigenvalue \lambda _1(M) of the Laplacian involved. [ABSTRACT FROM AUTHOR]

We develop a classification theory for real-analytic hypersurfaces in \mathbb {C}^{2} in the case when the hypersurface is of infinite type at the reference point. This is the remaining, not yet understood case in \mathbb {C}^{2} in the Problème local , formulated by H. Poincaré in 1907 and asking for a complete biholomorphic classification of real hypersurfaces in complex space. One novel aspect of our results is a notion of smooth normal forms for real-analytic hypersurfaces. We rely fundamentally on the recently developed CR-DS technique in CR-geometry. [ABSTRACT FROM AUTHOR]

We prove that the set of Segre-degenerate points of a real-analytic subvariety X in Cn is a closed semianalytic set. It is a subvariety if X is coherent. More precisely, the set of points where the germ of the Segre variety is of dimension k or greater is a closed semianalytic set in general, and for a coherent X, it is a real-analytic subvariety of X. For a hypersurface X in Cn, the set of Segre-degenerate points, X[n], is a semianalytic set of dimension at most 2n−4. If X is coherent, then X[n] is a complex subvariety of (complex) dimension n−2. Example hypersurfaces are given showing that X[n] need not be a subvariety and that it also need not be complex; X[n] can, for instance, be a real line. [ABSTRACT FROM AUTHOR]

We obtain L^2\times L^2\to L^1 boundedness for bilinear maximal functions associated with general compact hypersurfaces. Our method is based on the strategy introduced by Barrionuevo et al. [Math. Res. Lett. 25 (2018), pp. 69–1388] and a new multiplier result established by Grafakos, He, and Slavíková [Math. Ann. 376 (2020), pp. 431–455]. [ABSTRACT FROM AUTHOR]

We show by an elementary argument that the second fundamental form of a connected, totally umbilical hypersurface of class C^2 is a constant multiple of the metric tensor. It follows that the hypersurface is smooth and it is either a piece of a hyperplane or of a sphere. [ABSTRACT FROM AUTHOR]

We prove a mild strengthening of a theorem of C̆esnavic̆ius which gives a criterion for a vector bundle on a smooth complete intersection of dimension at least 3 to split into a sum of line bundles. We also prove an analogous statement for bundles on a general complete intersection surface. [ABSTRACT FROM AUTHOR]

For all integers k > 0, we prove that the hypergeometric function Îk(α)= ∑j=0√ ((8k + 4)j)!j!/(2j)!((2k + 1)j)!2((4k+1)j)! αj is a period of a pencil of curves of genus 3k + 1. We prove that the function Îk is a generating function of Gromov–Witten invariants of the family of anticanonical del Pezzo hypersurfaces X = X8k+4 ⊂ P(2,2k + 1, 2k + 1, 4k + 1). Thus, the pencil is a Landau–Ginzburg mirror of the family. The surfaces X were first constructed by Johnson and Kollár. The feature of these surfaces that makes our mirror construction especially interesting is that |−KX| = |OX (1)| = ∅. This means that there is no way to form a Calabi–Yau pair (X,D) out of X and hence there is no known mirror construction for X other than the one given here. We also discuss the connection between our construction and work of Beukers, Cohen and Mellit on hypergeometric functions. [ABSTRACT FROM AUTHOR]

We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this problem differs substantially from that of lacunary discrete maximal operators defined along a nonsingular hypersurface. Our positive results are improvements over bounds for the corresponding full maximal functions which were initially studied by Magyar. In order to obtain positive results, we use an interpolation technique of the second author to reduce problem to a maximal function of main terms. The main terms take the shape of those introduced in work of the first author, which is a more localized version of the main terms that appear in work of Magyar. The main ingredient of this paper is a new bound on the main terms near ℓ1. For our negative results we generalize an argument of Zienkiewicz. [ABSTRACT FROM AUTHOR]

We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every noncompact manifold admits a nonenlargeable metric. In proving the first result, we use the main result of the recent paper by Schoen and Yau on minimal hypersurfaces to obstruct positive scalar curvature in arbitrary dimensions. More concretely, we use this to study nonzero degree maps ƒ : Xn → Sk × Tn-k, with k = 1,2,3. When X is a closed oriented manifold endowed with a metric g of positive scalar curvature and the map ƒ is (possibly area) contracting, we prove inequalities relating the lower bound of the scalar curvature of g and the contracting factor of the map ƒ. [ABSTRACT FROM AUTHOR]

We present a quasi-local version of the stability of the positive mass theorem. We work with the Brown-York quasi-local mass as it possesses positivity and rigidity properties, and therefore the stability of this rigidity statement can be studied. Specifically, we ask if the Brown-York mass of the boundary of some compact manifold is close to zero, must the manifold be close to a Euclidean domain in some sense? Here we consider a class of compact n-manifolds with boundary that can be realized as graphs in Rn+1, and establish the following. If the Brown-York mass of the boundary of such a compact manifold is small, then the manifold is close to a Euclidean hyperplane with respect to the Federer-Fleming flat distance. [ABSTRACT FROM AUTHOR]

We prove a local version of a recently established theorem by Myroshnychenko, Ryabogin and the second named author. More specifically, we show that if n ≥ 3, g : Sn−1 → R is a bounded measurable function, U is an open connected subset of Sn−1 and the restriction (section) of ƒ onto any great sphere perpendicular to U is isotropic, then C(g)|U = c + < a,⋅ > and R(g)|U = c', for some fixed constants c,c' ∈ R and for some fixed vector a ∈ Rn. Here, C(g) denotes the cosine transform and R(g) denotes the Funk transform of g. However, we show that an even g does not need to be equal to a constant almost everywhere in U⊥ := ∪u∈U (Sn−1∩ u⊥). For the needs of our proofs, we obtain a new generalization of a result from classical differential geometry, in the setting of convex hypersurfaces, that we believe is of independent interest. [ABSTRACT FROM AUTHOR]

The paper is devoted to the description of those non-degenerate sets of points Z in P3 whose general projection to a general plane is a complete intersection of curves in that plane. One large class of such Z is what we call (a,b)-grids. We relate this problem to the unexpected cone property C(d), a special case of the unexpected hypersurfaces which have been the focus of much recent research. After an analysis of C(d) for small d, we show that a non-degenerate set of 9 points has a general projection that is the complete intersection of two cubics if and only if the points form a (3,3)-grid. However, in an appendix we describe a set of 24 points that are not a grid but nevertheless have the projection property. These points arise from the F4 root system. Furthermore, from this example we find subsets of 20, 16 and 12 points with the same feature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of expanding flows of closed, smooth, uniformly convex hypersurfaces in Euclidean Rn+1 with speed uα ƒβ (α, β ∈ R1), where u is the support function of the hypersurface, ƒ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If α ≤ 0 < β ≤ 1−α, we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalization, to a round sphere centered at the origin. [ABSTRACT FROM AUTHOR]

We show that the boundary of any bounded strongly pseudoconvex complete circular domain in C2 must contain points that are exceptionally tangent to a projective image of the unit sphere. [ABSTRACT FROM AUTHOR]

We show that the boundary of any bounded strongly pseudoconvex complete circular domain in C2 must contain points that are exceptionally tangent to a projective image of the unit sphere. [ABSTRACT FROM AUTHOR]

The purpose of this article is to prove that there exists a real smooth pseudoconvex hypersurface germ (M,p) of D'Angelo infinite type in Cn+1 such that it does not admit any (singular) holomorphic curve in Cn+1 tangent to M at p to infinite order. [ABSTRACT FROM AUTHOR]

The purpose of this article is to prove that there exists a real smooth pseudoconvex hypersurface germ (M,p) of D'Angelo infinite type in Cn+1 such that it does not admit any (singular) holomorphic curve in Cn+1 tangent to M at p to infinite order. [ABSTRACT FROM AUTHOR]