Theory Of Hypersurfaces Research Articles

Horizons that gyre and gimble: a differential characterization of null hypersurfaces

Motivated by the thermodynamics of black hole solutions conformal to stationary solutions, we study the geometric invariant theory of null hypersurfaces. It is well-known that a null hypersurface in a Lorentzian manifold can be treated as a Carrollian geometry. Additional structure can be added to this geometry by choosing a connection which yields a Carrollian manifold. In the literature various authors have introduced Koszul connections to study the study the physics on these hypersurfaces. In this paper we examine the various Carrollian geometries and their relationship to null hypersurface embeddings. We specify the geometric data required to construct a rigid Carrollian geometry, and we argue that a connection with torsion is the most natural object to study Carrollian manifolds. We then use this connection to develop a hypersurface calculus suitable for a study of intrinsic and extrinsic differential invariants on embedded null hypersurfaces; motivating examples are given, including geometric invariants preserved under conformal transformations.

Folded symplectic forms in contact topology

We establish the relationship between folded symplectic forms and convex hypersurface theory in contact topology. As an application, we use convex hypersurface theory to reprove and strengthen the existence result for folded symplectic forms due to Cannas da Silva, and we generalize to all even dimensions Baykur's 4-dimensional existence result of folded Weinstein structures and folded Lefschetz fibrations.

Nevanlinna theory for tropical hypersurfaces

Nevanlinna theory for tropical hypersurfaces

Unlocking multidimensional cancer therapeutics using geometric data science

Personalised approaches to cancer therapeutics primarily involve identification of patient sub-populations most likely to benefit from targeted drugs. Such a stratification has led to plethora of designs of clinical trials that are often too complex due to the need for incorporating biomarkers and tissue types. Many statistical methods have been developed to address these issues; however, by the time such methodology is available research in cancer has moved on to new challenges and therefore in order to avoid playing catch-up it is necessary to develop new analytic tools alongside. One of the challenges facing cancer therapy is to effectively and appropriately target multiple therapies for sensitive patient population based on a panel of biomarkers across multiple cancer types, and matched future trial designs. We present novel geometric methods (mathematical theory of hypersurfaces) to visualise complex cancer therapeutics data as multidimensional, as well as geometric representation of oncology trial design space in higher dimensions. The hypersurfaces are used to describe master protocols, with application to a specific example of a basket trial design for melanoma, and thus setup a framework for further incorporating multi-omics data as multidimensional therapeutics.

A stage structured predator-prey model with periodic orbits

<p style='text-indent:20px;'>This paper studies the existence and multiplicity of periodic orbits of a stage-structured predator-prey model with Beddington-DeAngelis functional response. We derive an existence criterion of periodic orbit in terms of inequalities of system's nine parameters and prove that the system admits at least two limit cycles or three limit cycles via subcritical Hopf bifurcation or generalized Hopf bifurcation theory and hypersurface theory. We also prove that each one-parameter system possesses at least one limit cycle when it is larger or smaller than its Hopf bifurcating value, or between its Hopf bifurcating values. Combining theoretic analysis and numerical simulations, we investigate global bifurcations of limit cycles for a varying parameter system, which provides a plenty of bifurcating properties and again suggests that the system has at least three limit cycles. Similar results are obtained for the system without mutual interference.</p>

Заметка о -квазиомбилических гиперплоскостях почти эрмитовых многообразий

In the present note, we consider the introduced by Lidia Vasil’evna Stepanova notion of an -quasi-umbilical hypersurface in an almost Her­mitian manifold. We show that the notion of an -quasi-umbilical hyper­surface in an almost Hermitian manifold is connected with the notion of a minimal hypersurface in this manifold. Using the classical theory of minimal hypersurfaces in Riemannian manifolds and Kirichenko — Stepanova general theory of almost contact metric hypersurfaces in almost Hermitian manifolds, we establish that an -quasi-umbilical hypersurface of a nearly Kählerian manifold is minimal if and only if this hypersurface is totally umbilical. Taking into account the connection between the notions of a minimal hypersurface and of an -quasi-umbilical hypersurface in an almost Her­mitian manifold, we conclude that some well-known results in the theory of almost contact metric hypersurfaces in almost Hermitian manifolds can be reformulated. The problem of the existence of a non-umbilical minimal -quasi-umbilical hypersurface of a quasi-Kählerian manifold is posed.

Translation Hypersurfaces in Euclidean 4-Spaces

In this article, the translation hypersurfaces in Euclidean 4- space are defined as the sum of three curves with distinct parameters with unit speed, and non-planar. These curves are called the generator curves of the hypersurface. Utilizing the hypersurface theory in Euclidean 4-space, unit normal vector field, shape (Weingarten) operator matrix, fundamental forms, Gaussian curvature and mean curvature have been expressed for the translation hypersurfaces. Finally, the computational example is stated to efficiency of the theoretical results.

Extremal transitions via quantum Serre duality

Two varieties Z and $${\widetilde{Z}}$$ are said to be related by extremal transition if there exists a degeneration from Z to a singular variety $${\overline{Z}}$$ and a crepant resolution $${\widetilde{Z}} \rightarrow {\overline{Z}}$$ . In this paper we compare the genus-zero Gromov–Witten theory of toric hypersurfaces related by extremal transitions arising from toric blow-up. We show that the quantum D-module of $${\widetilde{Z}}$$ , after analytic continuation and restriction of a parameter, recovers the quantum D-module of Z. The proof provides a geometric explanation for both the analytic continuation and restriction parameter appearing in the theorem.

Combinatorial study of stable categories of graded Cohen\u2013Macaulay modules over skew quadric hypersurfaces

In this paper, we present a new connection between representation theory of noncommutative hypersurfaces and combinatorics. Let S be a graded (pm 1)-skew polynomial algebra in n variables of degree 1 and f =x_1^2 + cdots +x_n^2 in S. We prove that the stable category mathsf {underline{CM}}^{mathbb Z}(S/(f)) of graded maximal Cohen–Macaulay module over S/(f) can be completely computed using the four graphical operations. As a consequence, mathsf {underline{CM}}^{mathbb Z}(S/(f)) is equivalent to the derived category mathsf {D}^{mathsf {b}}({mathsf {mod}},k^{2^r}), and this r is obtained as the nullity of a certain matrix over {mathbb F}_2. Using the properties of Stanley–Reisner ideals, we also show that the number of irreducible components of the point scheme of S that are isomorphic to {mathbb P}^1 is less than or equal to left( {begin{array}{c}r+1 2end{array}}right) .

Complex affine distributions

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.

SFT COMPUTATIONS AND INTERSECTION THEORY IN HIGHER-DIMENSIONAL CONTACT MANIFOLDS

AbstractI construct infinitely many nondiffeomorphic examples of $5$ -dimensional contact manifolds which are tight, admit no strong fillings and do not have Giroux torsion. I obtain obstruction results for symplectic cobordisms, for which I give a proof not relying on the polyfold abstract perturbation scheme for Symplectic Field Theory (SFT). These results are part of my PhD thesis [23], and are the first applications of higher-dimensional Siefring intersection theory for holomorphic curves and hypersurfaces, as outlined in [23, 24], as a prequel to [30].

Compact Dupin hypersurfaces

A hypersurface $M$ in ${\bf R}^n$ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if the number of distinct principal curvatures is constant on $M$, i.e., each continuous principal curvature function has constant multiplicity on $M$. These conditions are preserved by stereographic projection, so this theory is essentially the same for hypersurfaces in ${\bf R}^n$ or $S^n$. The theory of compact proper Dupin hypersurfaces in $S^n$ is closely related to the theory of isoparametric hypersurfaces in $S^n$, and many important results in this field concern relations between these two classes of hypersurfaces. In 1985, Cecil and Ryan conjectured on p. 184 of the book, Tight and Taut Immersions of Manifolds, that every compact, connected proper Dupin hypersurface $M \subset S^n$ is equivalent to an isoparametric hypersurface in $S^n$ by a Lie sphere transformation. This paper gives a survey of progress on this conjecture and related developments.

On the Multiplicity One Conjecture in min-max theory

We prove that in a closed manifold of dimension between $3$ and $7$ with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each volume spectrum is realized by the min-max value of certain relative homotopy class of sweepouts of boundaries of Caccioppoli sets. The main result follows by approximating such min-max value using the min-max theory for hypersurfaces with prescribed mean curvature established by the author with Zhu.

Equiaffine Braneworld

Higher dimensional theories, wherein our four dimensional universe is immersed into a bulk ambient, have received much attention recently, and the directions of investigation had, as far as we can discern, all followed the ordinary Euclidean hypersurface theory’s isometric immersion recipe, with the spacetime metric being induced by an ambient parent. We note, in this paper, that the indefinite signature of the Lorentzian metric perhaps hints at the lesser known equiaffine hypersurface theory as being a possibly more natural, i.e., less customized beyond minimal mathematical formalism, description of our universe’s extrinsic geometry. In this alternative, the ambient is deprived of a metric, and the spacetime metric becomes conformal to the second fundamental form of the ordinary theory, therefore is automatically indefinite for hyperbolic shapes. Herein, we advocate investigations in this direction by identifying some potential physical benefits to enlisting the help of equiaffine differential geometry. In particular, we show that a geometric origin for dark energy can be proposed within this framework.

1-Conformal geometry of quasi statistical manifolds

A quasi statistical manifold is a generalization of a statistical manifold. The notion of quasi statistical manifolds was introduced to formulate the geometry of non-conservative estimating functions in statistics. Later, it was showed that quasi statistical manifolds are induced from affine distributions in the same way as statistical manifolds are induced from affine immersions. Here, an affine distribution is a non-integrable version of an affine immersion, and it is useful in quantum information geometry. On the other hand, it is known that generalized conformal geometry is useful for the study of statistical manifolds from the viewpoint of affine differential geometry. In particular, 1-conformal geometry of statistical manifolds gives a relation with the notion of affine immersions. Although generalized conformal geometry of quasi statistical manifolds is also expected to be useful, the geometry has not been cleared yet. The aim of this paper is to formulate 1-conformal geometry of quasi statistical manifolds. We research a relation between 1-conformal geometry of quasi statistical manifolds and the notion of affine distributions. As the main result, we show the fundamental theorems for affine distributions. We also formulate a hypersurface theory of quasi statistical manifolds.

Einstein Hypersurfaces of $$\mathbb {S}^n \times \mathbb {R}$$ and $$\mathbb {H}^n \times \mathbb {R}$$

In this paper, we classify the Einstein hypersurfaces of $$\mathbb {S}^n \times \mathbb {R}$$ and $$\mathbb {H}^n \times \mathbb {R}$$ . We use the characterization of the hypersurfaces of $$\mathbb {S}^n \times \mathbb {R}$$ and $$\mathbb {H}^n \times \mathbb {R}$$ whose tangent component of the unit vector field spanning the factor $$\mathbb {R}$$ is a principal direction and the theory of isoparametric hypersurfaces of space forms to show that Einstein hypersurfaces of $$\mathbb {S}^n \times \mathbb {R}$$ and $$\mathbb {H}^n \times \mathbb {R}$$ must have constant sectional curvature.

Real Hypersurfaces in the Complex Hyperbolic Quadric with Reeb Parallel Structure Jacobi Operator

We introduce the notion of Reeb parallel structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$, $m \geq 3$, and give a classification theory for real hypersurfaces in ${{Q^*}^m}$, $m \geq 3$, with Reeb parallel structure Jacobi operator.

A calculus for conformal hypersurfaces and new higher Willmore energy functionals

Abstract 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].

Minimal surfaces and the Allen--Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates

The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show for generic metrics on a 3-manifold, minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen-Cahn setting, a strong form of the multiplicity one conjecture and the index lower bound conjecture of Marques-Neves in 3-dimensions regarding min-max constructions of minimal surfaces. Allen-Cahn min-max constructions were recently carried out by Guaraco and Gaspar-Guaraco. Our resolution of the multiplicity one and the index lower bound conjectures shows that these constructions can be applied to give a new proof of Yau's conjecture on infinitely many minimal surfaces in a 3-manifold with a generic metric (recently proven by Irie-Marques-Neves) with new geometric conclusions. Namely, we prove that a 3-manifold with a generic metric contains, for every $p$ = 1, 2, 3, ..., a two-sided embedded minimal surface with Morse index $p$ and area $p^{1/3}$, as conjectured by Marques-Neves.

Unexpected curves and Togliatti‐type surfaces

AbstractThe purpose of this note is to establish a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior. We identify unexpected plane curves of degree 4 as sections of a rational surface of degree 7 in with its osculating spaces of order 2 which in every point of have dimension lower than expected. We put this result in perspective with earlier examples of surfaces with defective osculating spaces due to Shifrin and Togliatti. Our considerations are rendered by an analysis of Lefschetz Properties of ideals associated with the studied surfaces.