Minimal Immersions Research Articles

Anti-browning action of melatonin is dependent on the type of tissue in minimally processed sweet potatoes

This study explored the impact of exogenous melatonin on oxidative damage, protective mechanisms against oxidation, and reactions leading to browning, as well as the accumulation of antioxidants in minimally processed white-fleshed and orange sweet potato roots. Sweet potato roots with white and orange pulp were subjected to minimal processing and immersion in melatonin solutions at concentrations of 0, 10, 100, and 1000 µmol L−1. Following packaging, these were stored at 5 ± 2 °C for 20 days. Results indicate that increasing melatonin concentrations significantly reduced browning symptoms in both white and orange sweet potato slices. This reduction was associated with decreased levels of hydrogen peroxide (H2O2), thiobarbituric acid-reactive substances (TBARs), and electrolyte leakage, indicating reduced cell membrane damage. Enhanced protection was observed through increased activity of antioxidant enzymes superoxide dismutase, catalase, and ascorbate peroxidase and higher levels of antioxidant metabolites, including vitamin C and phenolic compounds, as confirmed by in vitro assays for antioxidant capacity (FRAP and DPPH), however, the levels of total carotenoids, anthocyanins, and yellow flavonoids did not show a significant difference between the melatonin treatments. Notably, sweet potato slices treated with higher melatonin doses of 100 and 1000 µmol L−1 exhibited reduced phenylalanine ammonia-lyase activity and phenolic compound accumulation. This effect, coupled with lower polyphenol oxidase (PPO) and peroxidase (POD) activities, likely contributed to prolonged preservation quality by maintaining cellular redox homeostasis.

Minimal surfaces with symmetries

AbstractLet be a finite group acting on a connected open Riemann surface by holomorphic automorphisms and acting on a Euclidean space by orthogonal transformations. We identify a necessary and sufficient condition for the existence of a ‐equivariant conformal minimal immersion . We show in particular that such a map always exists if acts without fixed points on . Furthermore, every finite group arises in this way for some open Riemann surface and . We obtain an analogous result for minimal surfaces having complete ends with finite total Gaussian curvature, and for discrete groups acting on properly discontinuously and acting on by rigid transformations.

Classification of minimal immersions of conformally flat 3-tori and 4-tori into spheres by the first eigenfunctions

Classification of minimal immersions of conformally flat 3-tori and 4-tori into spheres by the first eigenfunctions

Investigating the characteristics of Clifford hypersurfaces and the unit sphere via a minimal immersion in $ S^{n+1} $

<p>In this article, we find the different sufficient conditions for a compact minimal hypersurface $ M $ of the unit sphere $ S^{n+1}, n\in \mathbb{Z}^{+} $ to be the Clifford hypersurface $ S^{\ell }(\sqrt{\frac{\ell }{n}})\times S^{m}(\sqrt{\frac{m}{n}}), $ where $ \ell, m\in \mathbb{Z}^{+}, \; \ell +m = n $ or the sphere $ S^{n} $. This classification is achieved by applying constraints to the tangent and normal components of the immersion.</p>

Contact CR $ \delta $-invariant: an optimal estimate for Sasakian statistical manifolds

<p>Chen (1993) developed the theory of $ \delta $-invariants to establish novel necessary conditions for a Riemannian manifold to allow a minimal isometric immersion into Euclidean space. Later, Siddiqui et al. (2024) derived optimal inequalities involving the CR $ \delta $-invariant for a generic statistical submanifold in a holomorphic statistical manifold of constant holomorphic sectional curvature. In this work, we extend the study of such optimal inequality to the contact CR $ \delta $-invariant on contact CR-submanifolds in Sasakian statistical manifolds of constant $ \phi $-sectional curvature. This paper concludes with a summary and final remarks.</p>

A characterization of homogeneous totally real minimal two-spheres in a complex hyperquadric

In this paper, we give a characterization of homogeneous totally real minimal two-spheres in a complex hyperquadric [Formula: see text]. Let [Formula: see text] be a totally real minimal immersion from two-sphere in [Formula: see text], and [Formula: see text] (see Sec. 2) are globally defined invariants relative to the first and second fundamental forms. We prove that if its Gauss curvature [Formula: see text] and [Formula: see text] are constants, and [Formula: see text] vanishes identically, then [Formula: see text] is congruent to [Formula: see text] constructed by the Boruvka spheres with [Formula: see text].

Extremal metrics for combinations of Laplace eigenvalues and minimal surfaces into ellipsoids

We give a variational method for existence and regularity of metrics which minimize combinations of eigenvalues of the Laplacian among metrics of unit area on a surface. We show that there are minimal immersions into ellipsoids parametrized by eigenvalues, such that the coordinate functions are eigenfunctions with respect to the minimal metrics. This work generalizes the author's maximization for one eigenvalue among metrics of unit area on a surface related to minimal surfaces into spheres. This work also generalizes the previous characterizations of critical metrics for one eigenvalue to any combination of eigenvalues from target spheres to target ellipsoids.

Minimal Kähler submanifolds in product of space forms

Abstract In this article, we study minimal isometric immersions of Kähler manifolds into product of two real space forms. We analyse the obstruction conditions to the existence of pluriharmonic isometric immersions of a Kähler manifold into those spaces and we prove that the only ones into 𝕊 m - 1 × ℝ {\mathbb{S}^{m-1}\times\mathbb{R}} and ℍ m - 1 × ℝ {\mathbb{H}^{m-1}\times\mathbb{R}} are the minimal isometric immersions of Riemannian surfaces. Furthermore, we show that the existence of a minimal isometric immersion of a Kähler manifold M 2 ⁢ n {M^{2n}} into 𝕊 m - 1 × ℝ {\mathbb{S}^{m-1}\times\mathbb{R}} and 𝕊 m - k × ℍ k {\mathbb{S}^{m-k}\times\mathbb{H}^{k}} imposes strong restrictions on the Ricci and scalar curvatures of M 2 ⁢ n {M^{2n}} . In this direction, we characterise some cases as either isometric immersions with parallel second fundamental form or anti-pluriharmonic isometric immersions.

Totally real flat minimal surfaces in the hyperquadric

AbstractIn this paper, we study geometry of totally real minimal surfaces in the complex hyperquadric , and obtain some characterizations of the harmonic sequence generated by these minimal immersions. For totally real flat surfaces that are minimal immersed in both and , we determine them for , and give a classification theorem when they are Clifford solutions.

On the connected components of the moduli space of equivariant minimal surfaces in pmb {mathbb{C}mathbb{H}}^2.

An equivariant minimal surface in mathbb{C}mathbb{H}^n is a minimal map of the Poincaré disc into mathbb{C}mathbb{H}^n which intertwines two actions of the fundamental group of a closed surface Sigma : a Fuchsian representation on the disc and an irreducible action by isometries on mathbb{C}mathbb{H}^n. The moduli space of these can been studied by relating it to the nilpotent cone in each moduli space of PU(n, 1)-Higgs bundles over the conformal surface corresponding to the map. By providing a necessary condition for points on this nilpotent cone to be smooth this article shows that away from the points corresponding to branched minimal immersions or ±-holomorphic immersions the moduli space is smooth. The argument is easily adapted to show that for mathbb{R}mathbb{H}^n the full space of (unbranched) immersions is smooth. For mathbb{C}mathbb{H}^2 we show that the connected components of the moduli space of minimal immersions are indexed by the Toledo invariant and the Euler number of the normal bundle of the immersion. This is achieved by studying the limit points of the mathbb {C}^times -action on the nilpotent cone. It is shown that the limit points as trightarrow 0 lead only to branched minimal immersions or ±-holomorphic immersions. In particular, the Euler number of the normal bundle can only jump by passing through branched minimal maps.

C-R Immersions and Sub-Riemannian Geometry

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form θ, we consider natural ϵ-contractions, i.e., contractions gϵM of the Levi form Gθ, such that the norm of the Reeb vector field T of (M, θ) is of order O(ϵ−1). We study isopseudohermitian (i.e., f∗Θ=θ) Cauchy–Riemann immersions f:M→(A,Θ) between strictly pseudoconvex CR manifolds M and A, where Θ is a contact form on A. For every contraction gϵA of the Levi form GΘ, we write the embedding equations for the immersion f:M→A,gϵA. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as ϵ→0+. For every isopseudohermitian immersion f:M→S2N+1 into a sphere S2N+1⊂CN+1, we show that Webster’s pseudohermitian scalar curvature R of (M, θ) satisfies the inequality R≤2n(f∗gΘ)(T,T)+n+1+12{∥H(f)∥gΘf2+∥traceGθΠH(M)∇⊤−∇∥f∗gΘ2} with equality if and only if B(f)=0 and ∇⊤=∇ on H(M)⊗H(M). This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms.

On the topology and index of minimal surfaces II

For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resolves several conjectures made by J. Choe and D. Hoffman concerning the classification of low-index minimal surfaces: we show that there is no complete two-sided immersed minimal surface in $\mathbb{R}^3$ of index two, complete embedded minimal surface with index three, or complete one-sided minimal immersion with index one.

Energy Estimates for the Tracefree Curvature of Willmore Surfaces and Applications

We prove an $$\varepsilon $$ -regularity result for the tracefree curvature of a Willmore surface with bounded second fundamental form. For such a surface, we obtain a pointwise control of the tracefree second fundamental form from a small control of its $$L^2$$ -norm. Several applications are investigated. Notably, we derive a gap statement for surfaces of the aforementioned type. We further apply our results to deduce regularity results for conformal minimal spacelike immersions into the de Sitter space $${\mathbb {S}}^{4,1}$$ .

On minimal immersions of a singular non-CSC extremal Kähler metric into 3-dimensional space forms

On minimal immersions of a singular non-CSC extremal Kähler metric into 3-dimensional space forms

Geometric Inequalities for a Submanifold Equipped with Distributions

The article introduces invariants of a Riemannian manifold related to the mutual curvature of several pairwise orthogonal subspaces of a tangent bundle. In the case of one-dimensional subspaces, this curvature is equal to half the scalar curvature of the subspace spanned by them, and in the case of complementary subspaces, this is the mixed scalar curvature. We compared our invariants with Chen invariants and proved geometric inequalities with intermediate mean curvature squared for a Riemannian submanifold. This gives sufficient conditions for the absence of minimal isometric immersions of Riemannian manifolds in a Euclidean space. As applications, geometric inequalities were obtained for isometric immersions of sub-Riemannian manifolds and Riemannian manifolds equipped with mutually orthogonal distributions.

Curvature Invariants of Equivariant Isometric Minimal Immersions into Grassmannian Manifolds

Using theories of principal fibre bundles and connections in [11], we study curvature invariants as Riemannian submanifolds for equivariant isometric minimal immersions from Riemannian or Hermitian symmetric spaces of compact type into Grassmannian manifolds. First we calculate principal curvatures of all equivariant full isometric minimal immersions from the complex projective line to complex quadrics. It is shown that they do not depend on the parameter of the moduli space. According to the preceding research, it is important for the study of curvature invariants to examine the behavior of higher order covariant derivative of the second fundamental form. It also enables us to distinguish those maps by curvature invariants in the moduli space. In addition, we calculate principal curvatures of all equivariant full holomorphic isometric embeddings between the complex projective spaces.

Non-positively curved Ricci Surfaces with catenoidal ends

A Ricci surface is defined to be a Riemannian surface \(({\varvec{M}},{\varvec{g}}_{\varvec{M}})\) whose Gauss curvature \({\varvec{K}}\) satisfies the differential equation \({\varvec{K}}\varvec{\Delta } {\varvec{K}} + {\varvec{g}}_{\varvec{M}}\left( {{\textbf {d}}{\varvec{K}}},{{\textbf {d}}{\varvec{K}}}\right) + {\textbf {4}}{\varvec{K}}^{\textbf {3}}={\textbf {0}}\). In the case where \({\varvec{K}}<{\textbf {0}}\), this equation is equivalent to the well-known Ricci condition for the existence of minimal immersions in \({\mathbb {R}}^3\). Recently, Andrei Moroianu and Sergiu Moroianu proved that a Ricci surface with non-positive Gauss curvature admits locally an isometric minimal immersion into \({\mathbb {R}}^3\). In this paper, we are interested in studying non-compact orientable Ricci surfaces with non-positive Gauss curvature. Firstly, we give a definition of catenoidal end for non-positively curved Ricci surfaces. Secondly, we develop a tool which can be regarded as an analogue of the Weierstrass data to obtain some classification results for non-positively curved Ricci surfaces of genus zero with catenoidal ends. Furthermore, we also give an existence result for non-positively curved Ricci surfaces of arbitrary positive genus which have finite catenoidal ends.

On conformal minimal immersions with constant curvature from two-spheres into the complex hyperquadrics

On conformal minimal immersions with constant curvature from two-spheres into the complex hyperquadrics

Flexible domains for minimal surfaces in Euclidean spaces

In this paper we introduce and investigate a new notion of flexibility for domains in Euclidean spaces Rn for n≥3 in terms of minimal surfaces which they contain. A domain Ω in Rn is said to be flexible if every conformal minimal immersion U→Ω from a Runge domain U in an open conformal surface M can be approximated uniformly on compacts, with interpolation on any given finite set, by conformal minimal immersion M→Ω. Together with hyperbolicity phenomena considered in recent works, this extends the dichotomy between flexibility and rigidity from complex analysis to minimal surface theory.

The Calabi–Yau problem for minimal surfaces with Cantor ends

We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in \mathbb{R}^3 with bounded image. The analogous result holds for holomorphic immersions into any complex manifold of dimension at least 2, for holomorphic null immersions into \mathbb{C}^n with n \geq 3 , for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for superminimal immersions into any selfdual or anti-self-dual Einstein four-manifold.