arXiv subjects

Zhenxiao Xie

Publications and source records attributed to Zhenxiao Xie.

15 recordsLinked to original sources

arXiv2026

On the Willmore energy of flat $n$-tori in $\mathbb{R}^N$ and Chen's conjecture for $n$-tori

This paper establishes the sharp lower bound $(4nπ^2)^{n/2}$ for the Willmore energy $\mathcal{W}$ of flat $n$-tori in the Euclidean space. Up to Möbius transformations, the Clifford $n$-torus $\mathbb{S}^1\bigl(\sqrt{1/n}\,\bigr) \times \cdots \times \mathbb{S}^1\bigl(\sqrt{1/n}\,\bigr) \subset \mathbb{S}^{2n-1} \subset \mathbb{R}^{2n}$ is shown to be the unique minimizer attaining this bound. This also confirms Chen's conjecture for flat $n$-tori. However, when $n \geq3 $, we show that Chen's conjecture fails on the total mean curvature of general immersed $n$-tori: certain Möbius transformations of the Clifford $n$-torus strictly decrease the total mean curvature.

math.DG↗

arXiv2026

Willmore surfaces in 4-dimensional conformal manifolds

This paper is dedicated to the exploration of the conformal Willmore functional for surfaces within 4-dimensional conformal manifolds. We provide a detailed calculation of both the first and second variations, and present the Euler-Lagrange equation of this functional in a conformally invariant form. Utilizing the second variation formula we derived, we demonstrate that the Clifford torus in $\mathbb{C}P^2$ is strictly Willmore-stable. This finding strongly supports the conjecture proposed by Montiel and Urbano [J. reine angew. Math. 546 2002, 139-154], which posits that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore functional among all tori. Moreover, by applying our formula to complex curves in $\mathbb{C}P^2$, we establish that the first nonzero eigenvalue of the Jacobi operator is at least 12. In the context of 4-dimensional locally symmetric spaces, we construct several holomorphic differentials to show that among all minimal 2-spheres, only those super-minimal ones can be Willmore.

math.DG↗

arXiv2026

A proof of the Willmore-type conjecture in $\mathbb{C}P^2$

In 2002, Montiel and Urbano conjectured that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore-type functional $ \mathcal{W}^-=\int_{T^2}(2+|H|^2)\,dA,$ either among all tori or among all Lagrangian tori. In this paper, we confirm this conjecture in the Lagrangian setting and disprove it in the general setting. We establish that every oriented closed Lagrangian surface of genus $g\geq 1$ in $\mathbb{C}P^2$ has $\mathcal{W}^-$-energy no less than that of the Clifford torus. Moreover, we construct non-Lagrangian deformations of the Clifford torus along which $\mathcal{W}^-$ strictly decreases.

math.DG↗

arXiv2026

Minimal isometric immersions of flat n-tori into spheres

In 1985, Bryant established that a flat $2$-torus admits a minimal isometric immersion into some round sphere if and only if a certain rationality condition is satisfied. We show that when $n\geq 3$, the rationality criterion is no longer a necessary, but a sufficient condition for a flat $n$-torus to admit minimal isometric immersions into spheres. We also derive an upper bound for the algebraic irrationality degree of such immersions. When $n=3$, this bound is sharp and explicit embedded examples are provided respectively for each possible degree. Moreover, by constructing a family of non-homogeneous minimal flat $3$-tori, we show that minimal isometric immersions (embeddings) of flat $n$-tori are not necessarily homogeneous when $n \geq 3$. In addition, we establish a deformation theorem that every flat $n$-torus admitting a minimal isometric spherical immersion can be isometrically, minimally and homogeneously immersed into a sphere of dimension at most $n^2+n-1$.

math.DG↗

arXiv2026

On the first eigenvalue of the area Jacobi operator for complex curves in Kähler surfaces

In this paper, we investigate the first eigenvalue $Λ_1$ of the area Jacobi operator for complex curves in Kähler surfaces, establishing an extrinsic counterpart to the classical Lichnerowicz theorem for the Laplace-Beltrami operator. By analyzing the second variation of a conformally invariant Willmore-type functional, we derive the lower bound $Λ_1 \geq 2\,\mathfrak{Ric}$, where $\mathfrak{Ric}$ denotes the infimum of the ambient Ricci curvature. For Kähler-Einstein surfaces with positive Einstein constant $\mathfrak{c}>0$, this bound reduces to $Λ_1 \geq 2\mathfrak{c}$. We then explore the equality case, computing the exact dimension of the corresponding first eigenspace in terms of the area, genus, and the dimension of a space of holomorphic sections. This analysis shows that the equality is achieved for all curves of genus $g \leq 1$.

math.DG↗

arXiv2026

Two classes of Willmore Surfaces in $\mathbb{S}^2\times \mathbb{S}^2$

We establish two classification theorems for Willmore surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$. Firstly, we prove that a Willmore surface which is also minimal must be either a special complex curve given by a slice or a diagonal; or, a minimal surface in a totally geodesic submanifold $\mathbb{S}^2 \times \mathbb{S}^1$ described by a solution of the sinh-Gordon equation in one variable. Secondly, we demonstrate that a Willmore surface is of product type if and only if it is the product of an elastic curve in $\mathbb{S}^2$ and a great circle.

math.DG↗

arXiv2025

Classification of sextic curves in the Fano 3-fold $\mathcal{V}_5$ with rational Galois covers in ${\mathbb P}^3$

In this paper, we classify sextic curves in the Fano $3$-fold $\bf \mathcal{V}_5$ (the smooth quintic del Pezzo $3$-fold) that admit rational Galois covers in the complex ${\mathbb P}^3$. We show that the moduli space of such sextic curves is of complex dimension $2$ through the invariants of the engaged Galois groups for the explicit constructions. This raises the intriguing question of understanding the moduli space of sextic curves in ${\mathcal V}_5$ through their Galois covers in ${\mathbb P}^3$.

math.AG↗

arXiv2024

Fano 3-folds and classification of constantly curved holomorphic $2$-spheres of degree $6$ in the complex Grassmannian $G(2,5)$

Up to now the only known constantly curved sextic curve, i.e., holomorphic 2-sphere of degree 6, in the complex $G(2,5)$ has been the first associated curve of the Veronese curve of degree 4, which indicates that such curves are rare to find. Exploring the rich interplay between the ramification of harmonic sequences in differential geometry and algebro-geometric properties of projectively equivalent Fano 3-folds of index 2 and degree 5, we invoke the moduli space structure of sextic curves in the Fano 3-fold often referred to as $V_5$ to confirm the rarity of constancy of curvature, by establishing that the harmonic sequence of a generic sextic curve in $G(2, 5)$ is totally unramified. This paper proposes to investigate from the Galois viewpoint the way ramification can appear in relation to the constancy of curvature among nongeneric sextic curves in $G(2, 5)$. We prove through elaborate $PSL_2$-transvectant and engaged unitary analyses that, up to the ambient unitary equivalence, the moduli space of constantly curved sextic curves in $G(2,5)$ that are $GL(5,{\mathbb C})$-equivalent to those in $V_5$ ramified at the $PSL_2$-invariant 1-dimensional singular locus somewhere, is semialgebraic of dimension 2 all members of which barring the above Veronese curve are nonhomogeneous. Many explicit examples can be constructed.

math.DG↗

arXiv2023

Classification of Minimal Immersions of Conformally Flat $3$-Tori and $4$-Tori in Spheres by The First Eigenfunctions

This paper is devoted to the study of minimal immersions of flat $n$-tori into spheres, especially those immersed by the first eigenfunctions (such immersion is called $λ_1$-minimal immersion), which also play important roles in spectral geometry. It is known that there are only two non-congruent $λ_1$-minimal $2$-tori in spheres, which are both flat. For higher dimensional case, the Clifford $n$-torus in $\mathbb{S}^{2n-1}$ might be the only known example in the literature. In this paper, by discussing the general construction of homogeneous minimal flat $n$-tori in spheres, we construct many new examples of $λ_1$-minimal flat $3$-tori and $4$-tori. In contrast to the rigidity in the case of $2$-tori, we show that there exists a $2$-parameter family of non-congruent $λ_1$-minimal flat $4$-tori. It turns out that the examples we constructed exhaust all $λ_1$-minimal immersions of conformally flat $3$-tori and $4$-tori in spheres. The classification involves some detailed investigations of shortest vectors in lattices, which can also be used to solve the Berger's problem on flat $3$-tori and $4$-tori. The dilation-invariant functional $λ_1(g)V(g)^{\frac{2}{n}}$ about the first eignvalue is proved to have maximal value among all flat $3$-tori and $4$-tori.

math.DG↗

arXiv2020

Structure of minimal 2-spheres of constant curvature in the complex hyperquadric

In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both $\mathbb{C}P^n$ and the hyperquadric of $\mathbb{C}P^n$. The moduli space of all those noncongruent ones is introduced, which can be described by certain complex symmetric matrices modulo an appropriate group action. Using this description, many examples, such as constantly curved holomorphic 2-spheres of higher degree, nonhomogenous minimal 2-spheres of constant curvature, etc., are constructed. Uniqueness is proven for the totally real constantly curved 2-sphere minimal in both the hyperquadric and $\mathbb{C}P^n$.

math.DG↗

arXiv2014

The Moebius geometry of Wintgen ideal submanifolds

Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. They are Moebius invariant objects. The mean curvature sphere defines a conformal Gauss map into a Grassmann manifold. We show that any Wintgen ideal submanifold has a Riemannian submersion structure over a Riemann surface with the fibers being round spheres. Then the conformal Gauss map is shown to be a super-conformal and harmonic map from the underlying Riemann surface. Some of our previous results are surveyed in the final part.

math.DG↗

arXiv2014

Wintgen ideal submanifolds of codimension two, complex curves, and Moebius geometry

Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. Using the framework of Moebius geometry, we show that in the codimension two case, the mean curvature sphere of the Wintgen ideal submanifold corresponds to an 1-isotropic holomorphic curve in a complex quadric Q. Conversely, any 1-isotropic complex curve in Q describes a 2-parameter family of m-dimensional spheres whose envelope is always a m-dimensional Wintgen ideal submanifold at the regular points. The relationship with Dajczer and Tojeiro's work on the same topic as well as the description in terms of minimal surfaces in the Euclidean space is also discussed.

math.DG↗

arXiv2014

Classification of Moebius homogeneous Wintgen ideal submanifolds

A submanifold in a real space form attaining equality in the DDVV inequality at every point is called a Wintgen ideal submanifold. They are invariant objects under the Moebius transformations. In this paper, we classify those Wintgen ideal submanifolds of dimension m>3 which are Moebius homogeneous. There are three classes of non-trivial examples, each related with a famous class of homogeneous minimal surfaces in $S^n$ or $CP^n$: the cones over the Veronese surfaces $S^2$ in $S^n$, the cones over homogeneous flat minimal surfaces in $S^n$, and the Hopf bundle over the Veronese embeddings of $CP^1$ in $CP^n$.

math.DG↗

arXiv2014

Moebius geometry of three dimensional Wintgen ideal submanifolds in S^5

Wintgen ideal submanifolds in space forms are those ones attaining equality at every point in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the normal scalar curvature. This property is conformal invariant; hence we study them in the framework of Moebius geometry, and restrict to three dimensional Wintgen ideal submanifolds in S^5. In particular we give Moebius characterizations for minimal ones among them, which are also known as (3-dimensional) austere submanifolds (in 5-dimensional space forms).

math.DG↗

arXiv2012

Chen-Gackstatter type surfaces in R^4_1: deformation, symmetry, and embeddedness

We find a 2-parameter family of deformations in R^4_1 of the classical Chen-Gackstatter surface explicitly, and show the existence of a larger 4-parameter family of deformations. Each of them still has genus one, a unique end, with total Gaussian curvature $-\int K=8π$. On the other hand, a uniqueness theorem is obtained when we assume that the surface has more than 4 symmetries. The problem of embeddedness is also discussed with some partial results.

math.DG↗