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"Manifolds (mathematics)"
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Existence of infinitely many minimal hypersurfaces in positive Ricci curvature
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min–max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.
Journal Article
Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature
In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension n≥3. For every bounded open subset Ω⊂M with smooth boundary, we prove that ∫∂ΩHn-1n-1dσ≥AVR(g)|Sn-1|,where H is the mean curvature of ∂Ω and AVR(g) is the asymptotic volume ratio of (M, g). Moreover, the equality holds true if and only if (M\\Ω,g) is isometric to a truncated cone over ∂Ω. An optimal version of Huisken’s Isoperimetric Inequality for 3-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue’s non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.
Journal Article
A Grassmann manifold handbook: basic geometry and computational aspects
The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point.
Journal Article
A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs
Conventional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) may incur in severe limitations when dealing with nonlinear time-dependent parametrized PDEs, as these are strongly anchored to the assumption of modal linear superimposition they are based on. For problems featuring coherent structures that propagate over time such as transport, wave, or convection-dominated phenomena, the RB method may yield inefficient reduced order models (ROMs) when very high levels of accuracy are required. To overcome this limitation, in this work, we propose a new nonlinear approach to set ROMs by exploiting deep learning (DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM, both the nonlinear trial manifold (corresponding to the set of basis functions in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to the projection stage in a linear ROM) are learned in a non-intrusive way by relying on DL algorithms; the latter are trained on a set of full order model (FOM) solutions obtained for different parameter values. We show how to construct a DL-ROM for both linear and nonlinear time-dependent parametrized PDEs. Moreover, we assess its accuracy and efficiency on different parametrized PDE problems. Numerical results indicate that DL-ROMs whose dimension is equal to the intrinsic dimensionality of the PDE solutions manifold are able to efficiently approximate the solution of parametrized PDEs, especially in cases for which a huge number of POD modes would have been necessary to achieve the same degree of accuracy.
Journal Article
Some Optimal Bounds for δ-Casorati Curvatures with Slant Factor in Trans-Sasakian Manifolds
In this article, we derive some optimal inequalities for slant submanifolds on trans-Sasakian manifolds coupled with quarter-symmetric non-metric connection (qsnmc), utilizing generalized normalized δ-Casorati curvatures. Additionally, we describe submanifolds for which the equality cases are true.
Journal Article
Hyperbolic ∗-Ricci Solitons and Gradient Hyperbolic ∗-Ricci Solitons on
In this research paper, we introduce the notions of hyperbolic ∗-Ricci solitons and gradient hyperbolic ∗-Ricci solitons. We study the hyperbolic ∗-Ricci solitons on a three-dimensional ε-trans-Sasakian manifold. Specifically, we determine the hyperbolic ∗-Ricci solitons on a three-dimensional (ε)-trans-Sasakian manifold with a conformal vector field and a proper ϕ(Q[sup.*])-type vector field. Using hyperbolic ∗-Ricci solitons with a conformal vector field, we discuss some geometric symmetries on a three-dimensional (ε)-trans-Sasakian. In addition, we exhibit the nature of gradient hyperbolic ∗-Ricci solitons on a three-dimensional (ε)-trans-Sasakian manifold endowed with a scalar concircular field. Moreover, we demonstrate an example on a three-dimensional (ε)-trans-Sasakian manifold that admits the hyperbolic ∗-Ricci solitons and find the rate of change of the hyperbolic ∗-Ricci solitons within the same example. Lastly, we also introduce the concept of modified second hyperbolic ∗-Ricci solitons.
Journal Article
A Study on the Behavior of Osculating and Rectifying Curves on Smooth Immersed Surfaces in Esup.3
This paper presents a detailed investigation into the isometric properties of osculating and rectifying curves on smooth immersed surfaces in E[sup.3] . We examine the geometric interactions between these curves, specifically when the osculating curve is associated with one surface and the rectifying curve with another. The main objective of this study is to identify the conditions under which these curves exhibit isometric behavior, preserving their intrinsic geometric properties along their respective Frenet frames. Our findings demonstrate that these curves retain isometric characteristics along the tangent, normal, and binormal directions, offering new insights into their structural invariance. This research makes a significant contribution to the broader field of differential geometry, with potential applications in surface theory.
Journal Article
A Morse-Bott Approach to Monopole Floer Homology and the Triangulation Conjecture
In the present work we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient
flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped equipped with a spin
eBook
Ricci Solitons on Weak beta-Kenmotsu f-Manifolds
Recent interest among geometers in f -structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by the author and R. Wolak as a generalization of Hermitian and Kähler structures, as well as f -structures, allow for a fresh perspective on the classical theory. In this paper, we study a new f -structure of this kind, called the weak β-Kenmotsu f -structure, as a generalization of K. Kenmotsu’s concept. We prove that a weak β-Kenmotsu f -manifold is a locally twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with β=const and equipped with an η-Ricci soliton structure whose potential vector field satisfies certain conditions are η-Einstein manifolds of constant scalar curvature.
Journal Article