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Linfeng Zhou

Publications and source records attributed to Linfeng Zhou.

15 recordsLinked to original sources
arXiv2026

The Rigidity of the closed three dimensional regular Landsberg metrics

The Landsberg Berwald conjecture asks whether every regular Landsberg metric is Berwald. We settle this conjecture for closed three-dimensional manifolds: every smooth strongly convex regular Landsberg metric on such a manifold is Berwald, without any reversibility assumption. Equivalently, there are no regular Landsberg "unicorns" on closed three-manifolds; this is a reformulation of the same conjecture, rather than a second independent conjecture.The proof combines a vanishing theorem for commuting Codazzi cubic tensors on closed surfaces, a rigidity theorem for three dimensional Minkowski norms with constant curvature indicatrices, and a rank one argument for the nonlinear curvature. The remaining R-quadratic case is settled by compactness along the geodesic flow. The fibrewise constant curvature theorem is an ingredient in this argument and does not assert the full Laugwitz conjecture.

math.DG↗
arXiv2026

ProactiveMobile: A Comprehensive Benchmark for Boosting Proactive Intelligence on Mobile Devices

Multimodal large language models (MLLMs) have made significant progress in mobile agent development, yet their capabilities are predominantly confined to a reactive paradigm, where they merely execute explicit user commands. The emerging paradigm of proactive intelligence, where agents autonomously anticipate needs and initiate actions, represents the next frontier for mobile agents. However, its development is critically bottlenecked by the lack of benchmarks that can address real-world complexity and enable objective, executable evaluation. To overcome these challenges, we introduce ProactiveMobile, a comprehensive benchmark designed to systematically advance research in this domain. ProactiveMobile formalizes the proactive task as inferring latent user intent across four dimensions of on-device contextual signals and generating an executable function sequence from a comprehensive function pool of 63 APIs. The benchmark features over 3,660 instances of 14 scenarios that embrace real-world complexity through multi-answer annotations. To ensure quality, a team of 30 experts conducts a final audit of the benchmark, verifying factual accuracy, logical consistency, and action feasibility, and correcting any non-compliant entries. Extensive experiments demonstrate that our fine-tuned Qwen2.5-VL-7B-Instruct achieves a success rate of 19.15%, outperforming o1 (15.71%) and GPT-5 (7.39%). This result indicates that proactivity is a critical competency widely lacking in current MLLMs, yet it is learnable, emphasizing the importance of the proposed benchmark for proactivity evaluation.

cs.AI↗
arXiv2025

A Weighted Llarull Type Theorem and its applications

This paper generalizes Llarull's classical scalar curvature rigidity theorem to the setting of weighted manifolds with P-scalar curvature. More precisely, we prove the refinement of Llarull's theorem for P-scalar curvature, which is similar to Listing's work \cite{listing2010scalar}. As an application, we establish a Llarull type theorem in the form of $\mathbb{S}^k\times\mathbb{T}^{n-k}$.

math.DG↗
arXiv2025

TEE-BFT: Pricing the Security of Data Center Execution Assurance

Blockchains face inherent limitations when communicating outside their own ecosystem, largely due to the Byzantine Fault Tolerant (BFT) 3f+1 security model. Trusted Execution Environments (TEEs) are a promising mitigation because they allow a single trusted broker to interface securely with external systems. This paper develops a cost-of-collusion principal-agent model for compromising a TEE in a Data Center Execution Assurance design. The model isolates the main drivers of attack profitability: a K-of-n coordination threshold, independent detection risk q, heterogeneous per-member sanctions F_i, and a short-window flow prize (omega) proportional to the value secured (beta times V). We derive closed-form deterrence thresholds and a conservative design bound (V_safe) that make collusion unprofitable under transparent parameter choices. Calibrations based on time-advantaged arbitrage indicate that plausible TEE parameters can protect on the order of one trillion dollars in value.

econ.TH↗
arXiv2022

Description of tempered exponential dichotomies by admissibility with no Lyapunov norms

Tempered exponential dichotomy formulates the nonuniform hyperbolicity for random dynamical systems. It was described by admissibility of a pair of function classes defined with Lyapunov norms, For MET-systems (systems satisfying the assumptions of multiplicative ergodic theorem (abbreviated as MET)), it can be described by admissibility of a pair without a Lyapunov norm. However, it is not known how to choose a suitable Lyapunov norms before a tempered exponential dichotomy is given, and there are examples of random systems which are not MET-systems but have a tempered exponential dichotomy. In this paper we give a description of tempered exponential dichotomy for general random systems, which may not be MET-systems, purely by measurable admissibility of three pairs of function classes with no Lyapunov norms. Further, restricting to the MET-systems, we obtain a simpler description of only one pair with no Lyapunov norms. Finally, we use our results to prove the roughness of tempered exponential dichotomies for parametric random systems and give a Hölder continuous dependence of the associated projections on the parameter.

math.DS↗
arXiv2021

A uniformization theorem in complex Finsler geometry

In complex Finsler geometry, an open problem is: does there exist a weakly Kähler Finsler metric which is not Kähler? In this paper, we give an affirmative answer to this open problem. More precisely, we construct a family of the weakly Kähler Finsler metrics which are non-Kähler. The examples belong to the unitary invariant complex Randers metrics. Furthermore, a uniformization theorem of the unitary invariant complex Randers metrics with constant holomorphic curvature is proved under the weakly Kähler condition.

math.DG↗
arXiv2014

The curvatures of spherically symmetric Finsler metrics in $R^n$

In this paper, we classify the spherically symmetric Berwald metrics in $\mathbb{R}^n$. For the spherically symmetric Landsberg metrics, we prove that there do not exist any non-Berwald metrics among the regular case. The partial differential equation systems which can respectively characterize the spherically symmetric Finsler metrics with constant flag curvature and Einstein metrics of this type is also obtained. Utilizing these equations, we find an effective way to construct the non-projective, non-Randers Finsler metrics with constant flag curvature and many explicit examples are given by this method.

math.DG↗
arXiv2014

The spherically symmetric Finsler metrics with isotropic S-curvature

In this paper, we investigate the spherically symmetric Finsler metrics with isotropic S-curvature and obtain a characterized equation. As an application, we prove that these metrics with Douglas type must be Randers metrics or Berwald metrics. This result leads to two classification theorems.

math.DG↗
arXiv2012

The Finsler surface with K=0 and J=0

In this short note, we verify R. Bryant's claim: there does exist the singular Landsberg Finsler surface with a vanishing flag curvature which is not Berwaldian.

math.DG↗
arXiv2011

Two dimensional disjoint minimal graphs

In this paper, under the assumption of Gauss curvature vanishing at infinity, we will prove Meeks' conjecture: the number of disjointly supported minimal graphs in $\mathbb{R}^3$ is at most two.

math.DG↗
arXiv2011

Gradient estimate for eigenforms of Hodge Laplacian

In this paper, we derive a gradient estimate for the linear combinations of eigenforms of the Hodge Laplacian on a closed manifold. The estimate is given in terms of the dimension, volume, diameter and curvature bound of the manifold. As an application, we obtain directly a sharp estimate for the heat kernel of the Hodge Laplacian.

math.DG↗
arXiv2010

Spherically symmetric Finsler metrics in R^n

In this paper, we give the general form of spherically symmetric Finsler metrics in $R^n$ and surprisedly find that many well-known Finsler metrics belong to this class. Then we explicitly express projective metrics of this type. The necessary and sufficient conditions that projective Finsler metrics with spherical symmetry have constant flag curvature are also obtained.

math.DG↗