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In this paper, we initiate the study of the multiplicative Euclidean submanifolds, exhibiting the multiplicative counterparts of the Gauss–Weingarten formulas. Some particular types such as multiplicative conic, spherical, and totally geodesic submanifolds are analyzed. In relation to such multiplicative submanifolds, we introduce multiplicative rectifying submanifolds and characterize by means of multiplicative concurrent vector fields. The multiplicative rectifying hypersurfaces are also obtained. [ABSTRACT FROM AUTHOR]

We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces with any given constant mean curvature 푐, and with smooth boundary contacting at any given constant angle 휃. Moreover, if 푐 is nonzero and 휃 is not π 2 , then our min-max solution always has multiplicity one. We also establish a stable Bernstein theorem for minimal hypersurfaces with certain contact angles in higher dimensions. [ABSTRACT FROM AUTHOR]

We study the behavior of minimal hypersurfaces in the Schwarzschild n-manifolds that intersect the horizon orthogonally along the boundary. We show that a free boundary minimal hypersurface and a totally geodesic hyperplane must intersect when the distance between them is achieved in a bounded region. We also discuss when the Schwarzschild metric is perturbed in a way that its scalar curvature is no longer positive. [ABSTRACT FROM AUTHOR]

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]

In this work, we study a hypersurface immersed in specific types of cylindrically symmetric static space-times, then we identify the gauge fields of the Lagrangian that minimizes the area beside the Noether symmetries. We show that these symmetries are part of the Killing algebra of cylindrically symmetric static space-times. By using Noether's theorem, we construct the conserved vector fields for the minimal hypersurface. [ABSTRACT FROM AUTHOR]

We prove that a |$C^{\infty }$| -generic area-preserving diffeomorphism of a closed, oriented surface admits a sequence of equidistributed periodic orbits. This is a quantitative refinement of the recently established generic density theorem for area-preserving surface diffeomorphisms. The proof has two ingredients. The first is a "Weyl law" for PFH spectral invariants, which was used to prove the generic density theorem. The second is a variational argument inspired by the work of Marques–Neves–Song and Irie on equidistribution results for minimal hypersurfaces and three-dimensional Reeb flows, respectively. [ABSTRACT FROM AUTHOR]

The space of degree d smooth projective hypersurfaces of |$\mathbb{C}P^n$| admits a scanning map to a certain space of sections. We compute a rational homotopy model of the action by conjugation of the group |$U(n+1)$| on this space of sections, from which we deduce that the scanning map induces a monomorphism on cohomology when d  > 2. Our main technique is the rational homotopy theory of Sullivan and, more specifically, a Sullivan model for the action by conjugation of a connected topological group on a space of sections. [ABSTRACT FROM AUTHOR]

The paper computes the Witt‐sheaf cohomology rings of partial flag varieties in type A in terms of the Pontryagin classes of the subquotient bundles. The proof is based on a Leray–Hirsch‐type theorem for Witt‐sheaf cohomology for the maximal rank cases, and a detailed study of cohomology ring presentations and annihilators of characteristic classes for the general case. The computations have consequences for the topology of real flag manifolds: we show that all torsion in the integral cohomology is 2‐torsion, which was not known in full generality previously. This allows for example to compute the Poincaré polynomials of complete flag varieties for cohomology with twisted integer coefficients. The computations also allow to describe the Chow–Witt rings of flag varieties, and we sketch an enumerative application to counting flags satisfying multiple incidence conditions to given hypersurfaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a volume- or area-preserving curvature flow for hypersurfaces with capillary boundary in the half-space, with speed given by a positive power of the mean curvature with a non-local averaging term. We demonstrate that for any convex initial hypersurface with a capillary boundary, the flow exists for all time and smoothly converges to a spherical cap as t → + ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we establish a Heintze–Karcher-type inequality for capillary hypersurfaces in a unit ball. To achieve this, we introduce a special Finsler metric given by Zermelo's navigation and study the geodesic normal flow with respect to this Finsler metric. Our results indicate that the relationship between capillary hypersufaces and hypersurfaces with free boundary is similar to the one between Finsler geometry and Riemannian geometry. [ABSTRACT FROM AUTHOR]

Let (A , m) be a Cohen–Macaulay local ring of dimension d ≥ 1 . Suppose there exists be a non-zero A module M of finite length and finite projective dimension such that ℓ ℓ (M) , the Lowey length of M, is equal to λ (M) , the length of M. Then we show that necessarily A is at worst a hypersurface singularity. We also characterize Gorenstein local rings having a non-zero module M of finite length and finite projective dimension with ℓ ℓ (M) = λ (M) - 1 . [ABSTRACT FROM AUTHOR]

We compute the dimension of certain components of the family of smooth determinantal degree d$d$ surfaces in P3${\mathbb {P}}^3$, and show that each of them is the closure of a component of the Noether–Lefschetz locus NL(d)$NL(d)$. Our computations exhibit that smooth determinantal surfaces in P3${\mathbb {P}}^3$ of degree 4 form a divisor in |OP3(4)|$|\mathcal {O}_{{\mathbb {P}}^3}(4)|$ with five irreducible components. We will compute the degrees of each of these components: 320,2508,136512,38475$320,2508,136512,38475$, and 320112$\hskip.001pt 320112$. [ABSTRACT FROM AUTHOR]

In this paper, we study centroids, orthocenters, circumcenters, and incenters of geodesic triangles in non-Euclidean geometry, and we discuss the existence of the Euler line in this context. Moreover, we give simple proofs of the existence of a totally geodesic 2-dimensional submanifold containing a given geodesic triangle in the hyperbolic or spherical 3-dimensional geometry. [ABSTRACT FROM AUTHOR]

The generalized Schwarzschild spacetimes are introduced as warped manifolds where the base is an open subset of R 2 equipped with a Lorentzian metric and the fiber is a Riemannian manifold. This family includes physically relevant spacetimes closely related to models of black holes. The generalized Schwarzschild spacetimes are endowed with involutive distributions which provide foliations by lightlike hypersurfaces. In this paper, we study spacelike submanifolds immersed in the generalized Schwarzschild spacetimes, mainly, under the assumption that such submanifolds lie in a leaf of the above foliations. In this scenario, we provide an explicit formula for the mean curvature vector field and establish relationships between the extrinsic and intrinsic geometry of the submanifolds. We have derived several characterizations of the slices, and we delve into the specific case where the warping function is the radial coordinate in detail. This subfamily includes the Schwarzschild and Reissner–Nordström spacetimes. [ABSTRACT FROM AUTHOR]

We determine a Simons type formula for spacelike submanifolds within a broad class of semi-Riemannian warped products, which includes the Robertson-Walker spacetimes. This formula enables us to extend results well-known from Riemannian geometry to the semi-Riemannian case. Specifically, when the ambient space has constant sectional curvature, such as Lorentz-Minkowski, de Sitter and anti-de Sitter spacetimes, we establish that compact spacelike hypersurfaces with parallel mean curvature vector field and non-negative sectional curvature are isoparametric hypersurfaces. This result constitutes a generalization of the Riemannian case within space forms, as demonstrated by Nomizu and Smyth in 1969. As an application, we extend analogous results previously established for pseudo-parallel immersions in Riemannian space forms. With this extension, we prove that any semi-parallel spacelike hypersurface with zero mean curvature into de Sitter spacetime is totally geodesic. Furthermore, we show that there are no semi-parallel spacelike hypersurfaces with zero mean curvature into Einstein-de Sitter spacetime. [ABSTRACT FROM AUTHOR]

In this paper, we study an open problem raised by Cecil and Ryan [Geometry of Hypersurfaces, Springer Monographs in Mathematics, p. 531] which asked whether there exist non‐Hopf Ricci‐pseudosymmetric hypersurfaces in CP2$\mathbb {C}P^{2}$ and CH2$\mathbb {C}H^{2}$. As our main results, we first prove the nonexistence of non‐Hopf Ricci‐pseudosymmetric hypersurfaces of the constant type in CH2$\mathbb {C}H^{2}$. Then, we prove the existence of non‐Hopf Ricci‐pseudosymmetric hypersurfaces of the constant type in CP2$\mathbb {C}P^{2}$. Finally, applying the preceding results and sharpening Theorem 4.1 of Wang and Zhang [J. Geom. Phys. 181 (2022), 104648], we prove the nonexistence of non‐Hopf weakly Einstein hypersurfaces with constant norm of Riemannian curvature tensor in both CP2$\mathbb {C}P^{2}$ and CH2$\mathbb {C}H^{2}$. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the well-known Kelvin–Nevanlinna–Royden criterion to check the p-hyperbolicity of the class of weighted manifolds for any p ∈ (1,∞). Besides, we introduce a new definition of a weighted cohomology and use it to derive a cohomological criterion for the weighted p-parabolicity. As a result, we acquire a global comparison principle for the weighted p-Laplacian and also receive a Liouville-type result for weighted p-harmonic functions. Moreover, we give a decay estimate for the weighted p-Laplacian on smooth metric measure spaces. Some applications to study the weighted p-parabolicity and f-minimal hypersurfaces are also given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce definitions for the integrated equivariant Milnor number μIG and the equivariant Milnor class ℳG(Z), for singular hypersurfaces. We prove that the μIG are constant on the strata in a Whitney stratification of Z, along with the correlation ℳG(Z) = ℳG,0(Z) = 1 |G|∑i=1kμ IG(x i) for hypersurfaces hosting isolated singularities x1,…,xk, where ℳG,0(Z) denotes the 0th equivariant Milnor class of Z.We also introduce the equivariant Fulton–Johnson class of singular hypersurfaces. We give an equivariant version of Verdier’s specialization morphism in homology, and also for constructible functions. This is used for finding a relation between equivariant Fulton–Johnson and Schwartz–MacPherson classes. [ABSTRACT FROM AUTHOR]

Let X ⊆ ℙn+1 be an integral hypersurface of degree d. We show that each locally Cohen–Macaulay instanton sheaf ℰ on X with respect to 풪X ⊗풪ℙn+1(1) in the sense of [V. Antonelli and G. Casnati, Instanton sheaves on projective schemes, J. Pure Appl. Algebra 227 (2023) 107246, Definition 1.3] yields the existence of Steiner bundles 풢 and ℱ on ℙn+1 of the same rank r and a morphism φ: 풢(−1) →ℱ∨ such that the form defining X to the power rk(ℰ) is exactly det(φ). We inspect several examples for low values of d, n and rk(ℰ). In particular, we show that the form defining a smooth integral surface in ℙ3 is the pfaffian of some skew-symmetric morphism φ: ℱ(−1) →ℱ∨, where ℱ is a suitable Steiner bundle on ℙ3 of sufficiently large even rank. [ABSTRACT FROM AUTHOR]

For any complete C -algebraic variety X and its associated compact C -analytic space X , it follows from the well known GAGA principle that the algebraic Picard group P i c (X) and the analytic Picard group P i c (X) are isomorphic. Our main purpose here is to investigate an analogous situation for non complete situation, namely C -algebraic affine hypersurfaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider stationary rotating cylindrically symmetric dust spacetimes. We first show that the Maitra spacetime is the unique nonrigidly (nonnull shear scalar) rotating solution with a regular axis and that is the most general one of the field equations. We are also able to demonstrate what Maitra’s paper does not show, where the solution is merely stated without any demonstration. Then we find that the nonrigidly rotating spacetimes, not necessarily regular, can be matched, across timelike cylindrical hypersurfaces, to a one-parameter family of stationary vacuum exteriors given by the Weyl class but not to the Lewis class, generalizing a result by Bonnor and Steidman. Among other properties, it is shown that the amount of rotating dust packed inside a cylinder is bigger if it is rigidly rotating than nonrigidly rotating. Finally, we show that in the nonrigidly rotating cases the fluid necessarily has vorticity and the spacetimes are not silent, where the magnetic part of the Weyl tensor is nonzero. [ABSTRACT FROM AUTHOR]

Let D λ d denote the family of monomial deformations of diagonal hypersurface over a finite field q given by D λ d : X 1 d + X 2 d + ⋯ + X n d = λ d X 1 h 1 X 2 h 2 ⋯ X n h n , where d , n ≥ 2 , h i ≥ 1 , ∑ i = 1 n h i = d , and gcd (d , h 1 , h 2 , ... , h n) = 1. The Dwork hypersurface is the case when d = n , that is, h 1 = h 2 = ⋯ = h n = 1. Formulas for the number of q -points on the Dwork hypersurfaces in terms of McCarthy's p-adic hypergeometric functions are known. In this paper, we provide a formula for the number of q -points on D λ d in terms of McCarthy's p-adic hypergeometric function which holds for d ≥ n. [ABSTRACT FROM AUTHOR]

We propose a notion of stability for constant k$k$‐mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem. [ABSTRACT FROM AUTHOR]

In this article, we introduce a mesoscale continuum model for membranes made of two different types of amphiphilic lipids. The model extends work by Peletier and the second author (Arch. Ration. Mech. Anal. 193, 2009) for the one‐phase case. We present a mathematical analysis of the asymptotic reduction to the macroscale when a key length parameter becomes arbitrarily small. We identify two main contributions in the energy: one that can be connected to bending of the overall structure and a second that describes the cost of the internal phase separations. We prove the Γ$$ \Gamma $$‐convergence towards a perimeter functional for the phase separation energy and construct, in two dimensions, recovery sequences for the convergence of the full energy towards a 2D reduction of the Jülicher–Lipowsky bending energy with a line tension contribution for phase separated hypersurfaces. [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]

This paper aims to present the theory of hypersurfaces for a novel class of manifolds called an indefinite (α, β)-type almost contact metric statistical manifold or trans-Sasakian statistical manifold with an (l,m)-type connection. The study delves into the analysis of screen semi-invariant lightlike hypersurfaces within this framework. Specific conditions on the recurrent and Lie recurrent structure tensor field have been established. Additionally, illustrative examples are provided to enhance the comprehension of the introduced concept. [ABSTRACT FROM AUTHOR]

We introduce a new invariant of G-varieties, the dual complex, which roughly measures how divisors in the complement of the free locus intersect. We show that the top homology group of this complex is an equivariant birational invariant of G-varieties. As an application, we demonstrate the non-linearizability of certain large abelian group actions on smooth hypersurfaces in projective space of any dimension and degree at least 3. [ABSTRACT FROM AUTHOR]

In this article, we derive a Schur-type Inequality in terms of the gradient $ r $-Almost Newton-Ricci-Yamabe soliton in $ (k, \mu) $-contact metric manifolds. We discuss the triviality for the compact gradient $ r $-Almost Newton-Ricci-Yamabe soliton in $ (k, \mu) $-Contact metric manifolds. In the end, we deduce a Schur-type inequality for the gradient $ r $-Almost Newton-Yamabe soliton in $ (k, \mu) $-contact metric manifolds, static Riemannian manifolds, and normal homogeneous compact Riemannian manifolds coupled with a projected Casimir operator.

We define the membership function of a set as the function that determines whether an input is an element of the set. Canetti, Rothblum, and Varia showed how to obfuscate evasive membership functions of hyperplanes over a finite field of order an exponentially large prime, assuming the hardness of a modified decisional Diffie-Hellman problem. Barak, Bitansky, Canetti, Kalai, Paneth, and Sahai extended their work from hyperplanes to hypersurfaces of bounded degree, assuming multilinear maps. Both works are limited to algebraic sets over large fields of prime orders, and are based on less standard assumptions, although they prove virtual black-box security.In this paper, we handle much more general algebraic sets based on more standard assumptions, and prove input-hiding security, which is not weaker nor stronger than virtual black-box security (i.e., they are incomparable). Our first obfuscator handles affine algebraic sets over finite fields of order an arbitrary prime power. It is based on the preimage-resistance property of cryptographic hash function families. Our second obfuscator applies to both affine and projective algebraic sets over finite fields of order a polynomial size prime power. It is based on the same hardness assumption(s) required by input-hiding small superset obfuscation. Our paper is the first to handle the obfuscation problem of projective algebraic sets over small finite fields. [ABSTRACT FROM AUTHOR]

We study the set of common F q –rational solutions of "smooth" systems of multivariate symmetric polynomials with coefficients in a finite field F q. We show that, under certain conditions, the set of common solutions of such polynomial systems over the algebraic closure of F q has a "good" geometric behavior. This allows us to obtain precise estimates on the corresponding number of common F q –rational solutions. In the case of hypersurfaces we are able to strengthen the results. We illustrate the interest of these estimates through their application to certain classical combinatorial problems over finite fields. [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]

In this paper, we study rigidity aspects of Penrose's singularity theorem. Specifically, we aim to answer the following question: if a spacetime satisfies the hypotheses of Penrose's singularity theorem except with weakly trapped surfaces instead of trapped surfaces, then what can be said about the global spacetime structure if the spacetime is null geodesically complete? In this setting, we show that we obtain a foliation of MOTS which generate totally geodesic null hypersurfaces. Depending on our starting assumptions, we obtain either local or global rigidity results. We apply our arguments to cosmological spacetimes (i.e., spacetimes with compact Cauchy surfaces) and scenarios involving topological censorship. [ABSTRACT FROM AUTHOR]

We study Calabi–Yau manifolds which are complete intersections of hypersurfaces of multidegree 1 in an m-fold product of n-dimensional projective spaces. Using the theory of Coxeter groups, we show that the birational automorphism group of such a Calabi–Yau manifold X is infinite and a free product of copies of Z . Moreover, we give an explicit description of the boundary of the movable cone Mov ¯ (X) . In the end, we consider examples for the general and non-general case and picture the movable cone and the fundamental domain for the action of Bir (X) . [ABSTRACT FROM AUTHOR]

Differential geometry of a kind of helical hypersurface family that depends on six parameters within the seven‐dimensional Euclidean space is explored. The curvatures of these hypersurfaces are determined, and their minimality is examined, along with illustrative examples being provided. Lastly, the Laplace–Beltrami operator for that kind of hypersurfaces is calculated. [ABSTRACT FROM AUTHOR]

AbstractWe calculate the Krull–Gabriel dimension of the functor category of the (stable) category of maximal Cohen–Macaulay modules over complete Cohen-Macaulay local rings of countable Cohen-Macaulay representation type. We show that the Krull–Gabriel dimension is 0 if and only if the ring is of finite Cohen–Macaulay representation type and that is 2 if the ring is a hypersurface of countable but not finite Cohen–Macaulay representation type. [ABSTRACT FROM AUTHOR]

Let d and n be positive integers, and E/F be a separable field extension of degree m = n + d n . We show that if | F | > 2 , then there exists a point P ∈ P n (E) which does not lie on any degree d hypersurface defined over F. In other words, the m Galois conjugates of P impose independent conditions on the m-dimensional F-vector space of degree d forms in x 0 , x 1 , ... , x n . As an application, we determine the maximal dimensions of linear systems L 1 and L 2 of hypersurfaces in P n over a finite field F, where every F-member of L 1 is reducible and every F-member of L 2 is irreducible. [ABSTRACT FROM AUTHOR]

In this paper, we discuss various analytical solutions of the planner geometry with anisotropic fluid as well as vanishing complexity factor condition in the framework of f(픾) theory. The matching conditions are employed at the interior and exterior hypersurfaces. The connection between the matter parameters and the Weyl tensor has been constructed. For dissipative and non-dissipative structures, we examine four different conformal Killing vector possibilities, while some possibilities adhere to the 풴TF = 0 condition along with the matching conditions, where 풴TF indicates the trace free part of the electric component of the Riemann tensor. Several theoretical models are provided to show how a non-static system evolves. [ABSTRACT FROM AUTHOR]

An induced additive action on a projective variety X ⊆ ℙn is a regular action of the group 픾am on X with an open orbit, which can be extended to a regular action on the ambient projective space ℙn. In this work, we classify all projective hypersurfaces admitting an induced additive action with a finite number of orbits. [ABSTRACT FROM AUTHOR]

Let X⊂Pn+1$X \subset \mathbb {P}^{n+1}$ be a smooth Fano hypersurface of dimension n$n$ and degree d$d$. The derived category of coherent sheaves on X$X$ contains an interesting subcategory called the Kuznetsov component AX$\mathcal {A}_X$. We show that this subcategory, together with a certain autoequivalence called the rotation functor, determines X$X$ uniquely if d>3$d > 3$ or if d=3$d = 3$ and n>3$n > 3$. This generalizes a result by Huybrechts and Rennemo, who proved the same statement under the additional assumption that d$d$ divides n+2$n+2$. [ABSTRACT FROM AUTHOR]

We consider subtleties of the horizon (null-hypersurface) limit in the Parikh-Wilczek Membrane Approach to Black Holes. Specifically, we refine the correspondence between the projected Einstein equations of gravity with matter and the Raychaudhuri-Damour-Navier-Stokes (RDNS) equations of relativistic hydrodynamics. For a general configuration of gravity with matter we obtain additional terms in the hydrodynamic equations, which include very specific combinations of the contracted logarithmic derivatives of a parameter (the regularization function) determining the proximity of a stretched membrane to the black hole horizon. Nevertheless, direct computations of the new terms for exact (Schwarzschild and Kerr) black hole solutions prompt the standard form of the RDNS equations, due to the non-expanding horizon property of these solutions. Therefore, the reduction of the extended RDNS equations to their classical form may be viewed as an additional consistency condition in the exact black hole solutions hydrodynamics, and may serve as a non-trivial test for various viable approximations of spacetime metrics. We compare in detail the Parikh-Wilczek Membrane Approach with the Gourgoulhon-Jaramillo method of a null-hypersurface description, as well as give the link of the obtained results to our previous work on the Kerr black holes. [ABSTRACT FROM AUTHOR]

In this paper, we study CR maps between hyperquadrics and Winkelmann hypersurfaces. Based on a previous study on the CR Ahlfors derivative of Lamel–Son and a recent result of Huang–Lu–Tang–Xiao on CR maps between hyperquadrics, we prove that a transversal CR map from a hyperquadric into a hyperquadric or a Winkelmann hypersurface extends to a local holomorphic isometric embedding with respect to certain Kähler metrics if and only if the Hermitian part of its CR Ahlfors derivative vanishes on an open set of the source. Our proof is based on relating the geometric rank of a CR map into a hyperquadric and its CR Ahlfors derivative. [ABSTRACT FROM AUTHOR]

This paper presents an elastoplastic model to estimate the hydromechanical behavior of unsaturated soils based on state boundary hypersurface. Through mechanical hypersurface, the influence of saturation on yield stress can be expressed in a full form rather than an incremental form. Two hydraulic hypersurfaces and one mechanical hypersurface are proposed to establish the model. Two hydraulic hypersurfaces, composed of degree of saturation, void ratio and matrix suction, define the plastic hydraulic boundary. The elastic hydraulic behavior of unsaturated soils can be represented by scanning lines between these two hydraulic hypersurfaces. The mechanical hypersurface, composed of degree of saturation, void ratio and effective stress, defines the plastic mechanical boundary. The elastic mechanical behavior of unsaturated soils can be represented by scanning lines below the mechanical hypersurfaces. A large number of laboratory tests are used to validated the proposed model, showing that it can reasonably capture important features of the hydromechanical behavior of unsaturated soils. [ABSTRACT FROM AUTHOR]

We study germs of real‐analytic Levi‐flat hypersurfaces with singularities on a boundary manifold. We prove the existence of a normal form for a real‐analytic Levi‐flat hypersurface which is defined by the vanishing of the real part of Bk$B_k$, Ck$C_k$, and F4$F_4$ singularities. Finally, we prove a version of Morse–Vey's lemma for a real‐analytic Levi‐flat hypersurface with singularities on the boundary. [ABSTRACT FROM AUTHOR]

Using a two‐dimensional version of the delta method, we establish an asymptotic formula for the number of rational points of bounded height on non‐singular complete intersections of cubic and quadric hypersurfaces of dimension at least 23 over Fq(t)$\mathbb {F}_q(t)$, provided char(Fq)>3$\operatorname{char}(\mathbb {F}_q)>3$. Under the same hypotheses, we also verify weak approximation. [ABSTRACT FROM AUTHOR]

Let F q n represent the finite field with q n elements. In this paper, our focus is on determining the number of F q n -rational points for two specific objects: an affine Artin–Schreier curve given by the equation y q - y = x (x q i - x) - λ , and an Artin–Schreier hypersurface given by the equation y q - y = ∑ j = 1 r a j x j (x j q i j - x j) - λ . Additionally, we establish that the Weil bound is only achieved in these cases when the trace of the element λ ∈ F q n over the subfield F q is equal to zero. [ABSTRACT FROM AUTHOR]

In this article, we find the different sufficient conditions for a compact minimal hypersurface M of the unit sphere S n + 1 , n ∈ Z + to be the Clifford hypersurface S ℓ (ℓ n) × S m (m n) , where ℓ , m ∈ Z + , ℓ + m = n or the sphere S n . This classification is achieved by applying constraints to the tangent and normal components of the immersion. [ABSTRACT FROM AUTHOR]

Suppose that |$\Sigma ^{n}\subset \mathbb{S}^{n+1}$| is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue |$\lambda _{1}$| of the induced Laplace–Beltrami operator on |$\Sigma $| satisfies |$\lambda _{1} \geq \frac{n}{2}+ a_{n}(\Lambda ^{6} + b_{n})^{-1}$|⁠ , where |$a_{n}$| and |$b_{n}$| are explicit dimensional constants and |$\Lambda $| is an upper bound for the length of the second fundamental form of |$\Sigma $|⁠. This provides the first explicitly computable improvement on Choi and Wang's lower bound |$\lambda _{1} \geq \frac{n}{2}$| without any further assumptions on |$\Sigma $|⁠. [ABSTRACT FROM AUTHOR]

We categorify the inclusion–exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods. [ABSTRACT FROM AUTHOR]

In this paper, we establish some modified defect relations for the Gauss map g$g$ of a complete minimal surface S⊂Rm$S\subset \mathbb {R}^m$ into a k$k$‐dimension projective subvariety V⊂Pn(C)(n=m−1)$V\subset \mathbb {P}^n(\mathbb {C})\ (n=m-1)$ with hypersurfaces Q1,...,Qq$Q_1,\ldots,Q_q$ of Pn(C)$\mathbb {P}^n(\mathbb {C})$ in N$N$‐subgeneral position with respect to V(N≥k)$V\ (N\ge k)$. In particular, we give the upper bound for the number q$q$ if the image g(S)$g(S)$ intersects each hypersurface Q1,...,Qq$Q_1,\ldots,Q_q$ a finite number of times and g$g$ is nondegenerate over Id(V)$I_d(V)$, where d=lcm(degQ1,...,degQq)$d=\text{lcm}(\deg Q_1,\ldots,\deg Q_q)$, that is, the image of g$g$ is not contained in any hypersurface Q$Q$ of degree d$d$ with V⊄Q$V\not\subset Q$. Our results extend and generalize the previous results for the case of the Gauss map and hyperplanes in a projective space. The results and the method of this paper have been applied by some authors to study the unicity problem of the Gauss maps sharing families of hypersurfaces. [ABSTRACT FROM AUTHOR]