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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]

For every complete and minimally immersed submanifold f : M n → S n + p whose second fundamental form satisfies | A | 2 ≤ n ⁢ p / (2 ⁢ p − 1) , we prove that it is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface in S 4 , thereby extending the well-known results by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete M n . We also obtain the corresponding result for complete hypersurfaces with non-vanishing constant mean curvature, due to Alencar & do Carmo in the compact case, under the optimal bound on the umbilicity tensor. In dimension n ≤ 6 , a pinching theorem for complete higher-codimensional submanifolds with non-vanishing parallel mean curvature is proved, partly generalizing previous work by Santos. Our approach is inspired by the conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni. [ABSTRACT FROM AUTHOR]

Let r ⩾ 3 r\geqslant 3 be an integer and 푄 any positive definite quadratic form in 푟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces S Q S_{Q} defined by x 3 = Q ⁢ ( y 1 , … , y r ) ⁢ z x^{3}=Q(y_{1},\dots,y_{r})z . This confirms Manin’s conjecture for any S Q S_{Q} . Our proof is based on analytic methods, and uses some estimates for character sums and moments of 퐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case r = 3 r=3 . [ABSTRACT FROM AUTHOR]

In this paper, we aim at identifying the level sets of the gauge norm in the Heisenberg group ℍ n via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in ℍ 1 under an assumption on the location of the singular set, and in ℍ n for n ≥ 2 in the proper class of horizontally umbilical hypersurfaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ (θ) ϕ ⁢ (ρ) α − 2 ⁢ ϕ ⁢ (ρ) 2 + | ∇ ¯ ⁢ ρ | 2 , a generalization of the Alexandrov problem ( α = n + 1 ) in hyperbolic space, where f ~ is a smooth positive function on S n , 휌 is the radial function of the hypersurface, ϕ ⁢ (ρ) = sinh ⁡ ρ and 퐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 . We also consider the cases 2 < α ≤ n + 1 under the evenness assumption of f ~ and prove the existence of solutions to the above equations. [ABSTRACT FROM AUTHOR]

In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space H n + 1 based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li ("An inverse curvature type hypersurface flow in H n + 1 ," (Preprint)) as follows (0.1) ∫ M λ ′ f 2 E 1 2 + | ∇ M f | 2 − ∫ M ∇ ̄ f λ ′ , ν + ∫ ∂ M f ≥ ω n 1 n ∫ M f n n − 1 n − 1 n provided that M is h-convex and f is a positive smooth function, where λ′(r) = coshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in ("A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold," Commun. Pure Appl. Math., vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the kth mean curvatures in H n + 1 by virtue of the Brendle-Guan-Li's flow ("An inverse curvature type hypersurface flow in H n + 1 ," (Preprint)) as below (0.2) ∫ M λ ′ f 2 E k 2 + | ∇ M f | 2 E k − 1 2 − ∫ M ∇ ̄ f λ ′ , ν ⋅ E k − 1 + ∫ ∂ M f ⋅ E k − 1 ≥ p k ◦ q 1 − 1 ( W 1 (Ω)) 1 n − k + 1 ∫ M f n − k + 1 n − k ⋅ E k − 1 n − k n − k + 1 provided that M is h-convex and Ω is the domain enclosed by M, pk(r) = ωn(λ′)k−1, W 1 (Ω) = 1 n | M | , λ′(r) = coshr, q 1 (r) = W 1 S r n + 1 , the area for a geodesic sphere of radius r, and q 1 − 1 is the inverse function of q1. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in ("Locally constrained curvature flows and geometric inequalities in hyperbolic space," Math. Ann., vol. 382, nos. 3–4, pp. 1425–1474, 2022). [ABSTRACT FROM AUTHOR]

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ (θ) ϕ ⁢ (ρ) α − 2 ⁢ ϕ ⁢ (ρ) 2 + | ∇ ¯ ⁢ ρ | 2 , a generalization of the Alexandrov problem ( α = n + 1 ) in hyperbolic space, where f ~ is a smooth positive function on S n , 휌 is the radial function of the hypersurface, ϕ ⁢ (ρ) = sinh ⁡ ρ and 퐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 . We also consider the cases 2 < α ≤ n + 1 under the evenness assumption of f ~ and prove the existence of solutions to the above equations. [ABSTRACT FROM AUTHOR]

We define and study a notion of minimal exponent for a local complete intersection subscheme 푍 of a smooth complex algebraic variety 푋, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara–Malgrange 푉-filtration associated to 푍. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology H Z r ⁢ (O X) , where 푟 is the codimension of 푍 in 푋. We also study its relation to the Bernstein–Sato polynomial of 푍. Our main result describes the minimal exponent of a higher codimension subscheme in terms of the invariant associated to a suitable hypersurface; this allows proving the main properties of this invariant by reduction to the codimension 1 case. A key ingredient for our main result is a description of the Kashiwara–Malgrange 푉-filtration associated to any ideal (f 1 , ... , f r) in terms of the microlocal 푉-filtration associated to the hypersurface defined by ∑ i = 1 r f i ⁢ y i . [ABSTRACT FROM AUTHOR]

We study the relation between isolated hypersurface singularities (e.g., ADE) and the quantum cohomology ring by using spectral invariants, which are symplectic measurements coming from Floer theory. We prove, under the assumption that the quantum cohomology ring is semi-simple, that (1) if the smooth Fano variety degenerates to a Fano variety with an isolated hypersurface singularity, then the singularity has to be an A m -singularity, (2) if the symplectic manifold contains an A m -configuration of Lagrangian spheres, then there are consequences for the Hofer geometry, and that (3) the Dehn twist reduces spectral invariants. [ABSTRACT FROM AUTHOR]

In this article, we establish some truncated second main theorems for holomorphic curves into projective spaces with some special hypersurfaces and give some applications. In addition, the defect relation, the algebraically degenerate conditions and uniqueness theorem for holomorphic curves with some special divisors may be improved.

Given a compact Riemannian manifold, of dimension between 3 and 7, with boundary, we adapt Song's covering method to the free boundary case to show that the Morse index of a free boundary minimal hypersurface grows linearly with the sum of its Betti numbers, where the constant of growth depends on an upper bound on the area of the free boundary minimal hypersurface in question. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we consider a partially overdetermined mixed boundary value problem in space forms and generalize the main result in [11] to the case of general domains with partial umbilical boundary in space forms. Precisely, we prove that a partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface. In the second part of this paper, we prove a Heintze–Karcher–Ros-type inequality for embedded hypersurfaces with free boundary lying on a horosphere or an equidistant hypersurface in the hyperbolic space. As an application, we show an Alexandrov-type theorem for constant mean curvature hypersurfaces with free boundary in these settings. [ABSTRACT FROM AUTHOR]

The almost contact metric structure that we have on a real hypersurface M in the complex quadric Qm = SOm+2/SOmSO2 allows us to define, for any nonnull real number k, the k-th generalized Tanaka-Webster connection on M, ∇ ^ (k) . Associated to this connection, we have Cho and torsion operators F X (k) and T X (k) , respectively, for any vector field X tangent to M. From them and for any symmetric operator B on M, we can consider two tensor fields of type (1,2) on M that we denote by B F (k) and B T (k) , respectively. We classify real hypersurfaces M in Qm for which any of those tensors identically vanishes, in the particular case of B being the structure Lie operator Lξ on M. [ABSTRACT FROM AUTHOR]

Let 픽 q be the finite field of odd characteristic p with q elements ( q = p n , n ∈ ℕ ) and let 픽 q * represent the set of nonzero elements of 픽 q . By making use of the Smith normal form of exponent matrices, we obtain an explicit formula for the number of rational points on the variety defined by the following system of equations over 픽 q : { ∑ i = 1 r a i (1) ⁢ x 1 e i ⁢ 1 (1) ⁢ ⋯ ⁢ x n e i ⁢ n (1) = b 1 , ∑ j ′ = 0 t - 1 ∑ i ′ = 1 r j ′ + 1 - r j ′ a r j ′ + i ′ (2) ⁢ x 1 e r j ′ + i ′ , 1 (2) ⁢ ⋯ ⁢ x n j ′ + 1 e r j ′ + i ′ , n j ′ + 1 (2) = b 2 , where b i ∈ 픽 q ( i = 1 , 2 ), t ∈ ℕ , 0 = n 0 < n 1 < n 2 < ⋯ < n t , n k - 1 < n ≤ n k for some 1 ≤ k ≤ t , 0 = r 0 < r 1 < r 2 < ⋯ < r t , a i (1) ∈ 픽 q * for i ∈ { 1 , ... , r } , a i ′ (2) ∈ 픽 q * for i ′ ∈ { 1 , ... , r t } , and the exponent of each variable is a positive integer. This generalizes the results obtained previously by Wolfmann, Sun, Cao, and others. Our result also gives a partial answer to an open problem raised by Hu, Hong and Zhao [S. N. Hu, S. F. Hong and W. Zhao, The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 2015, 135–153]. [ABSTRACT FROM AUTHOR]

We study a notion of relative entropy for certain hypersurfaces in hyperbolic space. We relate this quantity to the renormalized area introduced by Graham–Witten. We also obtain a monotonicity formula for relative entropy applied to mean curvature flows in hyperbolic space. [ABSTRACT FROM AUTHOR]

In this paper, we study area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below with Cheeger–Colding theory. Let N i be a sequence of smooth manifolds with Ricci curvature ≥ - n ⁢ κ 2 on B 1 + κ ′ ⁢ (p i) for constants κ ≥ 0 , κ ′ > 0 , and volume of B 1 ⁢ (p i) has a positive uniformly lower bound. Assume B 1 ⁢ (p i) converges to a metric ball B 1 ⁢ (p ∞) in the Gromov–Hausdorff sense. For a sequence of area-minimizing hypersurfaces M i in B 1 ⁢ (p i) with ∂ ⁡ M i ⊂ ∂ ⁡ B 1 ⁢ (p i) , we prove the continuity for the volume function of area-minimizing hypersurfaces equipped with the induced Hausdorff topology. In particular, each limit M ∞ of M i is area-minimizing in B 1 ⁢ (p ∞) provided B 1 ⁢ (p ∞) is a smooth Riemannian manifold. By blowing up argument, we get sharp dimensional estimates for the singular set of M ∞ in ℛ , and 풮 ∩ M ∞ . Here, ℛ and 풮 are the regular and singular parts of B 1 ⁢ (p ∞) , respectively. [ABSTRACT FROM AUTHOR]

We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly's identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ in a Cartan-Hadamard manifold M. In particular, we show that the first mean curvature integral of a convex hypersurface γ nested inside Γ cannot exceed that of Γ , which leads to a sharp lower bound for the total first mean curvature of Γ in terms of the volume it bounds in M in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ is parallel to Γ , or M has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds. [ABSTRACT FROM AUTHOR]

In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in ℝ n + 1 . First, using comparison geometry in the context of metric geometry, we derive explicit upper bounds for the entropy of all such self-shrinkers. Second, as an application we prove a smooth compactness theorem on the space of all such shrinkers. We also prove that there are only finitely many such self-shrinkers with an extra reflection symmetry. [ABSTRACT FROM AUTHOR]

We study the behaviour of rational curves tangent to a hypersurface under degenerations of the hypersurface. Working within the framework of logarithmic Gromov–Witten theory, we extend the degeneration formula to the logarithmically singular setting, producing a virtual class on the space of maps to the degenerate fibre. We then employ logarithmic deformation theory to express this class as an obstruction bundle integral over the moduli space of ordinary stable maps. This produces new refinements of the logarithmic Gromov–Witten invariants, encoding the degeneration behaviour of tangent curves. In the example of a smooth plane cubic degenerating to the toric boundary we employ localisation and tropical techniques to compute these refinements. Finally, we leverage these calculations to describe how embedded curves tangent to a smooth cubic degenerate as the cubic does; the results obtained are of a classical nature, but the proofs make essential use of logarithmic Gromov–Witten theory. [ABSTRACT FROM AUTHOR]

In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding's method in [T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 1997, 3, 477–501]. With Cheeger–Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space. As an application, if a sequence of minimal hypersurfaces converges to a metric cone C ⁢ Y × ℝ n - k ( 2 ≤ k ≤ n ) in a non-collapsing metric cone C ⁢ X × ℝ n - k obtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY. Namely, Y has only one connected component in X. [ABSTRACT FROM AUTHOR]

In this paper, Hopf hypersurfaces in a complex projective plane ℂP2(c) or a complex hyperbolic plane ℂH2(c) with constant scalar curvature are classified. For a non-Hopf hypersurface in ℂP2(c) with constant scalar curvature r, it is proved that if the structure vector field is an eigenvector of the Ricci operator, then either r = 7c/2 or r = 3c/2. Moreover, these two cases are determined completely under an additional condition. [ABSTRACT FROM AUTHOR]

A rank 5 vector bundle G on HT ht is constructed as the homology of a certain Beilinson monad. By the double point formula, HT ht = - 4. The entries of the third row are given by I i >= 2 I i - HT ht . Substituting this value in the double point formula, we get HT ht = -12. [Extracted from the article]

We provide qualitative bounds on the area and topology of separating constant mean curvature (CMC) surfaces of bounded (Morse) index. We also develop a suitable bubble-compactness theory for embedded CMC hypersurfaces with bounded index and area inside closed Riemannian manifolds in low dimensions. In particular, we show that convergence always occurs with multiplicity one, which implies that the minimal blow-ups (bubbles) are all catenoids. [ABSTRACT FROM AUTHOR]

In this paper, we investigate closed strictly convex hypersurfaces in ℝ n + 1 {\mathbb{R}^{n+1}} which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree 1 and inverse concave function of the principal curvatures with power greater than 1, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere 𝕊 + n + 1 {\mathbb{S}^{n+1}_{+}} with power greater than or equal to 1. [ABSTRACT FROM AUTHOR]

In this paper, we give the complete classification of real hypersurfaces in a nonflat complex space form from the viewpoint of the η-parallelism of the tensor field h(= (1/2)퓛ξϕ). In addition we investigate real hypersurfaces whose tensor h is either Killing type or transversally Killing tensor. In particular, we shall determine Hopf hypersurfaces whose tensor h is transversally Killing tensor by using an application of the classification of real hypersurfaces admitting η-parallelism with respect to the tensor h. [ABSTRACT FROM AUTHOR]

We prove that, for every closed (not necessarily convex) hypersurface Σ in ℝ n + 1 {\mathbb{R}^{n+1}} and every p > n {p>n} , the L p {L^{p}} -norm of the trace-free part of the anisotropic second fundamental form controls from above the W 2 , p {W^{2,p}} -closeness of Σ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime p ≤ n {p\leq n} , the lack of convexity assumptions may lead in general to bubbling phenomena. Moreover, we obtain a stability theorem for quasi-Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates in the convex setting. [ABSTRACT FROM AUTHOR]

We conclude the classification of cohomogeneity one actions on symmetric spaces of rank one by classifying cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. As a by-product of our proof, we produce uncountably many examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces. [ABSTRACT FROM AUTHOR]

In this paper, our purpose is to give a characterization of the generic special cubic fourfold which contains a smooth rational surface of degree 9 not homologous to a complete intersection. As corollaries, we will give an explicit construction of families of smooth surfaces in generic special cubic fourfolds X ∈ 𝒞δ for 6 < δ ≤ 30 and δ ≡ 0 (mod 6). This applies in particular to give an explicit construction of two different liaison class of smooth surfaces in all such special cubic fourfolds with the prescribed invariants. [ABSTRACT FROM AUTHOR]

We express the Mahler measures of 23 families of Laurent polynomials in terms of Eisenstein–Kronecker series. These Laurent polynomials arise as Landau–Ginzburg potentials on Fano 3-folds, sixteen of which define K3 hypersurfaces of generic Picard rank 19, and the rest are of generic Picard rank less than 19. We relate the Mahler measure at each rational singular moduli to the value at 3 of the L-function of some weight-3 newform. Moreover, we find ten exotic relations among the Mahler measures of these families. [ABSTRACT FROM AUTHOR]

We aim to classify the real hypersurfaces M in a Kaehler complex space form Mn(c) satisfying the two conditions φ l = l φ , $\varphi l=l\varphi ,$ where l = R (⋅ , ξ) ξ and φ $l=R(\cdot ,\xi)\xi \text{ and }\varphi $ is the almost contact metric structure of M, and (∇ ξ l) X = $\left({{\nabla }_{\xi }}l \right)X=$ ω(X)ξ, where where ω(X) is a 1-form and X is a vector field on M. These two conditions imply that M is a Hopf hypersurface and ω = 0. [ABSTRACT FROM AUTHOR]

The hybrid model is the Landau–Ginzburg-type theory that is expected, via the Landau–Ginzburg/ Calabi–Yau correspondence, to match the Gromov–Witten theory of a complete intersection in weighted projective space. We prove a wall-crossing formula exhibiting the dependence of the genus-zero hybrid model on its stability parameter, generalizing the work of [21] for quantum singularity theory and paralleling the work of Ciocan-Fontanine–Kim [7] for quasimaps. This completes the proof of the genus-zero Landau– Ginzburg/Calabi–Yau correspondence for complete intersections of hypersurfaces of the same degree, as well as the proof of the all-genus hybrid wall-crossing [11]. [ABSTRACT FROM AUTHOR]

We show how the topological and geometric properties of the family of null hypersurfaces in a Lorentzian manifold are related with the properties of the ambient manifold itself. In particular, we focus in how the presence of global symmetries and curvature conditions restrict the existence of compact null hypersurfaces. We use these results to show the influence on the existence of compact totally umbilic null hypersurfaceswhich are not totally geodesic. Finally we describe the restrictions that they impose in causality theory. [ABSTRACT FROM AUTHOR]

We give a formula for the Hodge numbers of a three-dimensional hypersurface in a weighted projective space with only ordinary triple points as singularities. [ABSTRACT FROM AUTHOR]

We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994. As time t → 0− the solutions collapse to a round point where 0 is the singular time. But as t → − ∞ the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions SJ × ℝn−J, 1 ≤ J ≤ n − 1. These results are the analog of the corresponding results in Ricci flow ( J = n − 1) and mean curvature flow. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean 𝑛-space. This flow involves 𝑘-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz–Christoffel–Minkowski problem. [ABSTRACT FROM AUTHOR]

Let M be a real hypersurface of a complex space form Mn(c) with c ≠ 0 and n ≥ 3. We show that the Ricci tensor S of M satisfies S(X, Y) = ag(X, Y) for all vector fields X and Y on the holomorphic distribution, a being a constant, if and only if M is a pseudo-Einstein real hypersurface. By doing this we can give the definition of pseudo-Einstein real hypersurface under weaker conditions. [ABSTRACT FROM AUTHOR]

Kaimakamis and Panagiotidou in [Taiwanese J. Math. 18(6) (2014), 1991–1998] proposed an open question: are there real hypersurfaces in nonflat complex space forms whose ∗-Ricci tensor satisfies the condition of 𝔻-parallelism? In this short note, we present an affirmative answer and prove that a three-dimensional real hypersurface in a nonflat complex space form has 𝔻-parallel ∗-Ricci tensor if and only if it is locally congruent to either a geodesic hypersphere of radius r in ℂ H2(c) with tanh ⁡(√|c|/2 r) = 1/2 or a ruled real hypersurface. [ABSTRACT FROM AUTHOR]

We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant nonnegative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifolds of nonnegative sectional curvature. Using a concept of "duality" for strictly convex hypersurfaces, we also obtain a new type of inequality, so-called "pseudo"-Harnack inequality, for expanding flows in the sphere and in the hyperbolic space. [ABSTRACT FROM AUTHOR]

We study classifying problems for real hypersurfaces in a complex two-plane Grassmannian G2(ℂm+2). In relation to the generalized Tanaka–Webster connection, we consider a new concept of parallel normal Jacobi operator for real hypersurfaces in G2(ℂm+2) and prove that a real hypersurface in G2(ℂm+2) with generalized Tanaka–Webster 𝔇⊥-parallel normal Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍPn in G2(ℂm+2), where m = 2n. [ABSTRACT FROM AUTHOR]

In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound, which generalize the celebrated Schoen–Simon–Yau interior curvature estimates up to the free boundary. Our curvature estimates imply a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For 3-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without a-priori area bound. This generalizes Schoen's interior curvature estimates to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi. [ABSTRACT FROM AUTHOR]

The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold. Recently it has been shown how, given a conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem [21]. This enables a route to proliferating conformal hypersurface invariants. The aim of this work is to give a self contained and explicit treatment of the calculus and identities required to use this machinery in practice. In addition we show how to compute the solution's asymptotics. We also develop the calculus for explicitly constructing the conformal hypersurface invariant differential operators discovered in [21] and in particular how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we prove that a class of energy functions proposed in a recent work have the right properties to be deemed higher-dimensional analogues of the Willmore energy. This complements recent progress on the existence and construction of different functionals in [22] and [20]. [ABSTRACT FROM AUTHOR]

We show short time existence for the evolution of triple junction clusters driven by the surface diffusion flow. On the triple line we use the boundary conditions derived by Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard equation with degenerated mobility. These conditions are concurrency of the triple junction, angle conditions between the hypersurfaces, continuity of the chemical potentials and a flux-balance. For the existence analysis we first write the geometric problem over a fixed reference surface and then use for the resulting analytic problem an approach in a parabolic Hölder setting. [ABSTRACT FROM AUTHOR]

Our purpose in this paper is to study complete linear Weingarten space-like hypersurface immersed in locally symmetric Lorentz space obeying some curvature conditions. Our approach is based on the use of a Simons type formula related to an appropriated Cheng-Yau modified operator jointly with some generalized maximum principles, we obtain that such a space-like hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. This result corresponds to a natural improvement of previous ones due to de Lima, dos Santos, Velásquez [On the umbilicity of complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space, São Paulo J. Math. Sci. 11 (2017), 456–470] and Alías, de Lima, dos Santos [New characterizations of linear Weingarten spacelike hypersurfaces in de Sitter space, Pacific J. Math. 292 (2018), 1–19]. [ABSTRACT FROM AUTHOR]

We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the "vanishing" time T: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter γ > 1 2 {\gamma>\frac{1}{2}} , there is a solution with the highest curvature blowing up at the rate (T - t) - (γ + 1 2) {(T-t)^{{-(\gamma+\frac{1}{2})}}}. (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the "Grim Reaper" solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ. [ABSTRACT FROM AUTHOR]

Given a closed complex hypersurface Z ⊂ ℂ N + 1 {Z\subset\mathbb{C}^{N+1}} ( N ∈ ℕ {N\in\mathbb{N}}) and a compact subset K ⊂ Z {K\subset Z} , we prove the existence of a pseudoconvex Runge domain D in Z such that K ⊂ D {K\subset D} and there is a complete proper holomorphic embedding from D into the unit ball of ℂ N + 1 {\mathbb{C}^{N+1}}. For N = 1 {N=1} , we derive the existence of complete properly embedded complex curves in the unit ball of ℂ 2 {\mathbb{C}^{2}} , with arbitrarily prescribed finite topology. In particular, there exist complete proper holomorphic embeddings of the unit disc 𝔻 ⊂ ℂ {\mathbb{D}\subset\mathbb{C}} into the unit ball of ℂ 2 {\mathbb{C}^{2}}. These are the first known examples of complete bounded embedded complex hypersurfaces in ℂ N + 1 {\mathbb{C}^{N+1}} with any control on the topology. [ABSTRACT FROM AUTHOR]

In this paper, we present the joint study of two main branches of topology, relative properties and hyperspaces, with the aim of giving an answer to Problem 3 in [Arhangeľskii, A. V.: A generic theorem in the theory of cardinal invariants of topological spaces, Comment. Math. Univ. Carolin. 36 (1995), 305–325]. [ABSTRACT FROM AUTHOR]

Let M be a real hypersurface in nonflat complex space forms of complex dimension two. In this paper, we prove that the shape operator of M is transversally Killing with respect to the generalized Tanaka-Webster connection if and only if M is locally congruent to a type (A) or (B) real hypersurface. We also prove that shape operator of M commutes with Cho operator on holomorphic distribution if and only if M is locally congruent to a ruled real hypersurface. [ABSTRACT FROM AUTHOR]