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Jianquan Ge

Publications and source records attributed to Jianquan Ge.

At least 37 records · Page 2Linked to original sources
arXiv2026

Normal Scalar Curvature Inequality on a Class of Austere Submanifolds

In this paper, we establish new normal scalar curvature inequalities on a class of austere submanifolds by proving sharper DDVV-type inequalities on associated austere subspaces. We also provide some examples of austere submanifolds in this class and point out one of them achieves the equality in our normal scalar curvature inequality everywhere. As a byproduct, we obtain a Simons-type gap theorem for closed austere submanifolds in unit spheres which belong to that class.

math.DG↗
arXiv2025

Complete hypersurfaces in $R^{n+1}$ with constant mean and scalar curvature

In this paper, we investigate the rigidity problems of complete hypersurfaces with constant mean curvature and constant scalar curvature in Euclidean spaces. Firstly, under some conditions of Gaussian-Kronecker curvature, we provide characterizations for the unsolved cases of Núñez's theorems in dimensions 4 and 5, as well as several rigidity results under some conditions of $r$-th mean curvatures. Moreover, for the case of dimension 6, we also present analogous rigidity results. Finally, for general dimensions, we offer a rigidity theorem under similar pinching conditions.

math.DG↗
arXiv2025

Chern Conjecture on Minimal Willmore Hypersurfaces with Constant Scalar Curvature

In this paper, we prove that for an $n$-dimensional closed minimal Willmore hypersurface $M^n$ with constant scalar curvature in the unit sphere $\mathbb{S}^{n+1}$, the squared norm $S$ of the second fundamental form of $M^n$ satisfies $S\geqslant n+\frac{4n+9-\sqrt{4 n^{2}+60 n+81}}{2}$ if $S>n$. This proves, in the approximate sense, the Chern conjecture about the second gap ($S\geqslant 2n$ if $S>n$), which will be fully verified under a further inequality condition about the 4-th mean curvature.

math.DG↗
arXiv2025

Integral-Einstein hypersurfaces in spheres

Combining the intrinsic and extrinsic geometry, we generalize Einstein manifolds to Integral-Einstein (IE) submanifolds. A Takahashi-type theorem is established to characterize minimal hypersurfaces with constant scalar curvature (CSC) in unit spheres which are conjectured to be isoparametric in the Chern conjecture. For these hypersurfaces, we obtain some integral inequalities with the bounds characterizing exactly the totally geodesic hypersphere, the non-IE minimal Clifford torus $S^{1}(\sqrt{\frac{1}{n}})\times S^{n-1}(\sqrt{\frac{n-1}{n}})$ and the IE minimal CSC hypersurfaces. Moreover, if further the third mean curvature is constant, then it is an IE hypersurface or an isoparametric hypersurface with $g\leq2$ principal curvatures. In particular, all minimal isoparametric hypersurfaces with $g\geq3$ principal curvatures are IE hypersurfaces. As applications, we obtain some spherical Bernstein theorems, including that any embedded closed minimal surface of genus no more than $\mathfrak{g}$ inside a tubular neighborhood of constant radius $r(\mathfrak{g})>0$ around an equator in $\mathbb{S}^3$ is an equator.

math.DG↗
arXiv2025

Austere Matrices, Austere Submanifolds and Dupin Hypersurfaces

Motivated by Bryant's research on austere subspaces and Cartan's isoparametric hypersurfaces with 3 distinct principal curvatures, we construct three families of austere submanifolds with flat normal bundle in unit spheres. From these examples we find three irreducible proper Dupin hypersurfaces with 5 distinct principal curvatures of different multiplicities. Thus, we give a negative answer to an open question raised by Thorbergsson in 2000 which is instructive for the local classification of proper Dupin hypersurfaces. Moreover, as an application, we obtain an upper bound estimate for the dimension of austere subspaces.

math.DG↗
arXiv2023

Holomorphic d-scalar curvature on almost Hermitian manifolds

In this paper, we study the existence of constant holomorphic d-scalar curvature and the prescribing holomorphic d-scalar curvature problem on closed, connected almost Hermitian manifolds of dimension $n\geq6$. In addition, we obtain an application and a variation formula for the associated conformal invariant.

math.DG↗
arXiv2022

Isoparametric polynomials and sums of squares

Hilbert's 17th problem asks that whether every nonnegative polynomial can be a sum of squares of rational functions. It has been answered affirmatively by Artin. However, the question as to whether a given nonnegative polynomial is a sum of squares of polynomials is still a central question in real algebraic geometry. In this paper, we solve this question completely for the nonnegative polynomials associated with isoparametric polynomials, initiated by E. Cartan, which define the focal submanifolds of the corresponding isoparametric hypersurfaces.

math.DG↗
arXiv2022

Cohomology vanishing theorems for free boundary submanifolds

In this paper, via a new Hardy type inequality, we establish some cohomology vanishing theorems for free boundary compact submanifolds $M^n$ with $n\geq2$ immersed in the Euclidean unit ball $\mathbb{B}^{n+k}$ under one of the pinching conditions $|Φ|^2\leq C$, $|A|^2\leq \widetilde{C}$, or $|Φ|\leq R(p,|H|)$, where $A$ $(Φ)$ is the (traceless) second fundamental form, $H$ is the mean curvature, $C,\widetilde{C}$ are positive constants and $R(p,|H|)$ is a positive function. In particular, we remove the condition on the flatness of the normal bundle, solving the first question, and partially answer the second question on optimal pinching constants proposed by Cavalcante, Mendes and Vitório.

math.DG↗
arXiv2021

A lower bound for $L_2$ length of second fundamental form on minimal hypersurfaces

We prove a weak version of the Perdomo Conjecture, namely, there is a positive constant $δ(n)>0$ depending only on $n$ such that on any closed embedded, non-totally geodesic, minimal hypersurface $M^n$ in $\mathbb{S}^{n+1}$, $$\int_{M}S \geq δ(n){\rm Vol}(M^n),$$ where $S$ is the squared length of the second fundamental form of $M^n$. The Perdomo Conjecture asserts that $δ(n)=n$ which is still open in general. As byproducts, we also obtain some integral inequalities and Simons-type pinching results on closed embedded (or immersed) minimal hypersurfaces, with the first positive eigenvalue $λ_1(M)$ of the Laplacian involved.

math.DG↗
arXiv2021

On a Conjecture of Bahri-Xu

In order to study the Yamabe changing-sign problem, Bahri and Xu proposed a conjecture which is a universal inequality for $p$ points in $\mathbb R^m$. They have verified the conjecture for $p\leq3$. In this paper, we first simplify this conjecture by giving two sufficient and necessary conditions inductively. Then we prove the conjecture for the basic case $m=1$ with arbitrary $p$. In addition, for the cases when $p=4,5$ and $m\geq2$, we manage to reduce them to the basic case $m=1$ and thus prove them as well.

math.CA↗
arXiv2020

Some generalizations of the DDVV and BW inequalities

In this paper we generalize the known DDVV-type inequalities for real (skew-)symmetric and complex (skew-)Hermitian matrices to arbitrary real, complex and quaternionic matrices. Inspired by the Erdős-Mordell inequality, we establish the DDVV-type inequalities for matrices in the subspaces spanned by a Clifford system or a Clifford algebra. We also generalize the Böttcher-Wenzel inequality to quaternionic matrices.

math.DG↗