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Xianfeng Wang

Publications and source records attributed to Xianfeng Wang.

18 recordsLinked to original sources
arXiv2026

Hypersurfaces of $\mathbb{H}^2\times\mathbb{H}^2$ with constant sectional curvature

In this paper, we classify the hypersurfaces of $\mathbb{H}^2\times\mathbb{H}^2$ with constant sectional curvature. In contrast to $\mathbb{S}^2\times\mathbb{S}^2$, the resulting examples for $\mathbb{H}^2\times\mathbb{H}^2$ exhibit more diversity, and we construct a special example with non-constant product angle function. For $\mathbb{S}^2\times\mathbb{S}^2$, however, the product angle function of any constant sectional curvature hypersurface is identically zero. As a byproduct, we classify the hypersurfaces of $\mathbb{H}^2\times\mathbb{H}^2$ with constant product angle function and constant mean curvature (or constant scalar curvature).

math.DG↗
arXiv2026

LifeEval: A Multimodal Benchmark for Assistive AI in Egocentric Daily Life Tasks

The rapid progress of Multimodal Large Language Models (MLLMs) marks a significant step toward artificial general intelligence, offering great potential for augmenting human capabilities. However, their ability to provide effective assistance in dynamic, real-world environments remains largely underexplored. Existing video benchmarks predominantly assess passive understanding through retrospective analysis or isolated perception tasks, failing to capture the interactive and adaptive nature of real-time user assistance. To bridge this gap, we introduce LifeEval, a multimodal benchmark designed to evaluate real-time, task-oriented human-AI collaboration in daily life from an egocentric perspective. LifeEval emphasizes three key aspects: task-oriented holistic evaluation, egocentric real-time perception from continuous first-person streams, and human-assistant collaborative interaction through natural dialogues. Constructed via a rigorous annotation pipeline, the benchmark comprises 4,075 high-quality question-answer pairs across 6 core capability dimensions. Extensive evaluations of 26 state-of-the-art MLLMs on LifeEval reveal substantial challenges in achieving timely, effective and adaptive interaction, highlighting essential directions for advancing human-centered interactive intelligence.

cs.AI↗
arXiv2026

KidVis: Do Multimodal Large Language Models Possess the Visual Perceptual Capabilities of a 6-Year-Old?

While Multimodal Large Language Models (MLLMs) have demonstrated impressive proficiency in high-level reasoning tasks, such as complex diagrammatic interpretation, it remains an open question whether they possess the fundamental visual primitives comparable to human intuition. To investigate this, we introduce KidVis, a novel benchmark grounded in the theory of human visual development. KidVis deconstructs visual intelligence into six atomic capabilities - Concentration, Tracking, Discrimination, Memory, Spatial, and Closure - already possessed by 6-7 year old children, comprising 10 categories of low-semantic-dependent visual tasks. Evaluating 20 state-of-the-art MLLMs against a human physiological baseline reveals a stark performance disparity. Results indicate that while human children achieve a near-perfect average score of 95.32, the state-of-the-art GPT-5 attains only 67.33. Crucially, we observe a "Scaling Law Paradox": simply increasing model parameters fails to yield linear improvements in these foundational visual capabilities. This study confirms that current MLLMs, despite their reasoning prowess, lack the essential physiological perceptual primitives required for generalized visual intelligence.

cs.CV↗
arXiv2023

Generalized Alexandrov theorems in spacetimes with integral conditions

We investigate integral conditions involving the mean curvature vector $\vec{H}$ or mixed higher-order mean curvatures, to determine when a codimension-two submanifold $Σ$ lies on a shear-free (umbilical) null hypersurface in a spacetime. We generalize the Alexandrov-type theorems in spacetime introduced in \cite{wang2017Minkowski} by relaxing the curvature conditions on $Σ$ in several aspects. Specifically, we provide a necessary and sufficient condition, in terms of a mean curvature integral inequality, for $Σ$ to lie in a shear-free null hypersurface. A key component of our approach is the use of Minkowski formulas with arbitrary weight, which enables us to derive rigidity results for submanifolds with significantly weaker integral curvature conditions.

math.DG↗
arXiv2023

Shifted inverse curvature flows in hyperbolic space

We introduce the shifted inverse curvature flow in hyperbolic space. This is a family of hypersurfaces in hyperbolic space expanding by $F^{-p}$ with positive power $p$ for a smooth, symmetric, strictly increasing and $1$-homogeneous curvature function $f$ of the shifted principal curvatures with some concavity properties. We study the maximal existence and asymptotical behavior of the flow for horo-convex hypersurfaces. In particular, for $0<p\leq 1$ we show that the limiting shape of the solution is always round as the maximal existence time is approached. This is in contrast to the asymptotical behavior of the (non-shifted) inverse curvature flow, as Hung and Wang [18] constructed a counterexample to show that the limiting shape of inverse curvature flow in hyperbolic space is not necessarily round.

math.DG↗
arXiv2023

Hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$ with constant sectional curvature

In this paper, we classify the hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$ with constant sectional curvature. By applying the so-called Tsinghua principle, which was first discovered by the first three authors in 2013 at Tsinghua University, we prove that the constant sectional curvature can only be $\frac{1}{2}$ and the product angle function $C$ defined by Urbano is identically zero. We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in $\mathbb{S}^2\times\mathbb{S}^2$ with $C=0$, and we establish a one-to-one correspondence between the involving minimal hypersurface and the famous ``sinh-Gordon equation'' $$ (\frac{\partial^2}{\partial u^2}+\frac{\partial^2}{\partial v^2})h =-\tfrac{1}{\sqrt{2}}\sinh(\sqrt{2}h). $$ As a byproduct, we give a complete classification of the hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$ with constant mean curvature and constant product angle function $C$.

math.DG↗
arXiv2021

Self-similar solutions to fully nonlinear curvature flows by high powers of curvature

In this paper, we investigate closed strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree $1$ and inverse concave function of the principal curvatures with power greater than $1$, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere $\mathbb{S}^{n+1}_{+}$ with power greater than or equal to $1$.

math.DG↗
arXiv2019

Minimal Lagrangian submanifolds of the complex hyperquadric

We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding the geometry of the Lagrangian submanifold at hand. We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface. We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all, respectively all but one, local angle functions coincide.

math.DG↗
arXiv2019

Contracting axially symmetric hypersurfaces by powers of the $σ_k$-curvature

In this paper, we investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in $\mathbb{R}^{n+1}$ and $\mathbb{S}^{n+1}$ by $σ_k^α$, where $σ_k$ is the $k$-th elementary symmetric function of the principal curvatures and $α\ge 1/k$. We prove that for any $n\geq3$ and any fixed $k$ with $1\leq k\leq n$, there exists a constant $c(n,k)>1/k$ such that that if $α$ lies in the interval $[1/k,c(n,k)]$, then we have a nice curvature pinching estimate involving the ratio of the biggest principal curvature to the smallest principal curvature of the flow hypersurface, and we prove that the properly rescaled hypersurfaces converge exponentially to the unit sphere. In the case $1<k\le n \le k^2$, we can choose $c(n,k)=\frac{1}{k-1}$. Our results provide an evidence for the general convergence result without initial curvature pinching conditions.

math.DG↗
arXiv2018

Surfaces expanding by non-concave curvature functions

In this paper, we first investigate the flow of convex surfaces in the space form $\mathbb{R}^3(κ)~(κ=0,1,-1)$ expanding by $F^{-α}$, where $F$ is a smooth, symmetric, increasing and homogeneous of degree one function of the principal curvatures of the surfaces and the power $α\in(0,1]$ for $κ=0,-1$ and $α=1$ for $κ=1$. By deriving that the pinching ratio of the flow surface $M_t$ is no greater than that of the initial surface $M_0$, we prove the long time existence and the convergence of the flow. No concavity assumption of $F$ is required. We also show that for the flow in $\mathbb{H}^3$ with $α\in (0,1)$, the limit shape may not be necessarily round after rescaling.

math.DG↗
arXiv2018

Sharp Reilly-type inequalities for submanifolds in space forms

Let $M$ be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form $\mathbb{R}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on $M$ defined by $L_{T}f=-div(T\nabla f)$, where $T$ is a general symmetric, positive definite and divergence-free $(1,1)$-tensor on $M$. The upper bound is given in terms of an integration involving $tr T$ and $|H_T|^2$, where $tr T$ is the trace of the tensor $T$ and $H_T=\sum_{i=1}^nA(Te_i,e_i)$ is a normal vector field associated with $T$ and the second fundamental form $A$ of $M$. Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous `Reilly inequality'. The operator $L_{T}$ can be regarded as a natural generalization of the well-known operator $L_r$ which is the linearized operator of the first variation of the $(r+1)$-th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of Grosjean ([16]) and Li-Wang ([19]) for hypersurfaces to higher codimension case.

math.DG↗
arXiv2016

New characterizations of the Clifford torus as a Lagrangian self-shrinker

In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus $\mathbb{S}^1(1)\times\mathbb{S}^1(1)$ is the unique compact orientable Lagrangian self-shrinker in $\mathbb{C}^2$ with $|A|^2\leq 2$, which gives an affirmative answer to Castro-Lerma's conjecture. We also prove that the Clifford torus is the unique compact orientable embedded Lagrangian self-shrinker with nonnegative or nonpositive Gauss curvature in $\mathbb{C}^2$.

math.DG↗
arXiv2016

Lagrangian submanifolds in the nearly kaehler $s^3 \times s^3$

In this paper, we investigate Lagrangian submanifolds in the nearly Kaehler $S^3 \times S^3$. We construct a new example which is a at Lagrangian torus. We give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature in the nearly Kaehler $S^3\times S^3$.

math.DG↗
arXiv2013

Lagrangian submanifolds in complex space forms satisfying an improved equality involving $δ(2,2)$

It was proved in [8,9] that every Lagrangian submanifold $M$ of a complex space form $\tilde M^{5}(4c)$ of constant holomorphic sectional curvature $4c$ satisfies the following optimal inequality: {align}\tag{A}δ(2,2)\leq \text{\small${25}{4}$} H^{2}+8c,{align} where $H^{2}$ is the squared mean curvature and $δ(2,2)$ is a $δ$-invariant on $M$ introduced by the first author. This optimal inequality improves a special case of an earlier inequality obtained in [B.-Y. Chen, Japan. J. Math. 26 (2000), 105-127]. The main purpose of this paper is to classify Lagrangian submanifolds of $\tilde M^{5}(4c)$ satisfying the equality case of the improved inequality (A).

math.DG↗
arXiv2012

On stable compact minimal submanifolds of Riemannian product manifolds

In this paper, we prove a classification theorem for the stable compact minimal submanifolds of the Riemannian product of an $m_1$-dimensional ($m_1\geq3$) hypersurface $M_1$ in the Euclidean space and any Riemannian manifold $M_2$, when the sectional curvature $K_{M_1}$ of $M_1$ satisfies $\frac{1}{\sqrt{m_1-1}}\leq K_{M_1}\leq 1.$ This gives a generalization to the results of F. Torralbo and F. Urbano [9], where they obtained a classification theorem for the stable minimal submanifolds of the Riemannian product of a sphere and any Riemannian manifold. In particular, when the ambient space is an $m$-dimensional ($m\geq3$) complete hypersurface $M$ in the Euclidean space, if the sectional curvature $K_{M}$ of $M$ satisfies $\frac{1}{\sqrt{m+1}}\leq K_{M}\leq 1$, then we conclude that there exist no stable compact minimal submanifolds in $M$.

math.DG↗
arXiv2010

Second eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature

Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere $\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral $\int_MH dv$ of the mean curvature $H$. In this paper, we derive an optimal upper bound for the second eigenvalue of the Jacobi operator $J_s$ of $M$. Moreover, when $r>1$, the bound is attained if and only if $M$ is totally umbilical and non-totally geodesic, when $r=1$, the bound is attained if $M$ is the Riemannian product $\mathbb{S}^{m}(c)\times\mathbb{S}^{n-m}(\sqrt{1-c^2}),~1\leq m\leq n-2,~c=\sqrt{\frac{(n-1)m+\sqrt{(n-1)m(n-m)}}{n(n-1)}}$.

math.DG↗
arXiv2010

Calabi product Lagrangian immersions in complex projective space and complex hyperbolic space

Starting from two Lagrangian immersions and a Legendre curve $\tildeγ(t)$ in $\mathbb{S}^3(1)$ (or in $\mathbb{H}_1^3(1)$), it is possible to construct a new Lagrangian immersion in $\mathbb{CP}^n$ (or in $\mathbb{CH}^n$), which is called a warped product Lagrangian immersion. When $\tildeγ(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, r_2e^{i(- \frac{r_1}{r_2}at)})$ (or $\tildeγ(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, r_2e^{i(\frac{r_1}{r_2}at)})$), where $r_1$, $r_2$, and $a$ are positive constants with $r_1^2+r_2^2=1$ (or $-r_1^2+r_2^2=-1$), we call the new Lagrangian immersion a Calabi product Lagrangian immersion. In this paper, we study the inverse problem: how to determine from the properties of the second fundamental form whether a given Lagrangian immersion of $\mathbb{CP}^n$ or $\mathbb{CH}^n$ is a Calabi product Lagrangian immersion. When the Calabi product is minimal, or is Hamiltonian minimal, or has parallel second fundamental form, we give some further characterizations.

math.DG↗