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Hanbing Fang

Publications and source records attributed to Hanbing Fang.

7 recordsLinked to original sources
arXiv2026

Strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow

In this paper, we extend the results of \cite{fang2025strong, fang2025singular} to generalized cylinders. More precisely, we establish a Lojasiewicz inequality for the pointed $\mathcal{W}$-entropy in Ricci flow under the assumption that the geometry near the base point is close to a generalized cylinder $\mathbb{R}^k \times N^{n-k}$, where $N$ is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition. As an application, we prove the strong uniqueness of generalized cylindrical tangent flows. Furthermore, we show that the subset $\mathcal{S}^k_{\mathrm{qc}}(N)\subset \mathcal{S}^k$, consisting of points at which some tangent flow is given by $\mathbb{R}^k \times N^{n-k}$ or its quotient, is horizontally parabolic $k$-rectifiable.

math.DG↗
arXiv2025

Singular sets in noncollapsed Ricci flow limit spaces

In this paper, we study the singular set $\mathcal{S}$ of a noncollapsed Ricci flow limit space, arising as the pointed Gromov--Hausdorff limit of a sequence of closed Ricci flows with uniformly bounded entropy. The singular set $\mathcal{S}$ admits a natural stratification: \begin{equation*} \mathcal S^0 \subset \mathcal S^1 \subset \cdots \subset \mathcal S^{n-2}=\mathcal S, \end{equation*} where a point $z \in \mathcal S^k$ if and only if no tangent flow at $z$ is $(k+1)$-symmetric. In general, the Hausdorff dimension of $\mathcal S^k$ with respect to the spacetime distance is at most $k$. We show that the subset $\mathcal{S}^k_{\mathrm{qc}} \subset \mathcal{S}^k$, consisting of points where some tangent flow is given by a standard cylinder or its quotient, is parabolic $k$-rectifiable. In dimension four, we prove the stronger statement that each stratum $\mathcal{S}^k$ is parabolic $k$-rectifiable for $k \in \{0, 1, 2\}$. Furthermore, we establish a sharp uniform $\mathscr{H}^2$-volume bound for $\mathcal{S}$ and show that, up to a set of $\mathscr{H}^2$-measure zero, the tangent flow at any point in $\mathcal{S}$ is backward unique. In addition, we derive $L^1$-curvature bounds for four-dimensional closed Ricci flows. As an application, we resolve Perelman's bounded diameter conjecture for three-dimensional closed Ricci flows.

math.DG↗
arXiv2025

Strong uniqueness of tangent flows at cylindrical singularities in Ricci flow

In this paper, we establish a Lojasiewicz inequality for the pointed $\mathcal{W}$-entropy in the Ricci flow, under the assumption that the geometry near the base point is close to a standard cylinder $\mathbb{R}^k \times S^{n-k}$ or the quotient thereof. As an application, we prove the strong uniqueness of the cylindrical tangent flow at the first singular time of the Ricci flow. Specifically, we show that the modified Ricci flow near the singularity converges to the cylindrical model under a fixed gauge.

math.DG↗
arXiv2025

On the structure of noncollapsed Ricci flow limit spaces

We establish a weak compactness theorem for the moduli space of closed Ricci flows, each equipped with a natural spacetime distance, under pointed Gromov--Hausdorff convergence. For the subspace of flows with uniformly bounded entropy, we further develop a structure theory for the corresponding noncollapsed Ricci flow limit spaces, showing that the regular part, where convergence is smooth, admits the structure of a Ricci flow spacetime, while the singular set has codimension at least four.

math.DG↗
arXiv2020

Phase Transitions in Germanium Telluride Nanoparticle Phase-Change Materials Studied by Time-Resolved X-Ray Diffraction

Germanium telluride (GeTe), a phase-change material, is known to exhibit four different structural phases: three at room temperature (one amorphous and two crystalline, $\alpha$ and $\gamma$) and one at high temperature (crystalline $\beta$). Because transitions between the amorphous and crystalline phases lead to significant changes in material properties (e.g., refractive index and resistivity), GeTe has been investigated as a phase-change material for photonics, thermoelectrics, ferroelectrics, and spintronics. Consequently, the temperature-dependent phase transitions in GeTe have been studied for bulk and thin-film GeTe, both fabricated by sputtering. Colloidal synthesis of nanoparticles offers a more flexible fabrication approach for amorphous and crystalline GeTe. These nanoparticles are known to exhibit size-dependent properties, such as an increased crystallization temperature for the amorphous-to-$\alpha$ transition in sub-10\,nm GeTe particles. The $\alpha$-to-$\beta$ phase transition is also expected to vary with size, but this effect has not yet been investigated for GeTe. Here, we report time-resolved X-ray diffraction of GeTe nanoparticles with different diameters and from different synthetic protocols. We observe a non-volatile amorphous-to-$\alpha$ transition between 210$^{\circ}$C and 240$^{\circ}$C and a volatile $\alpha$-to-$\beta$ transition between 370$^{\circ}$C and 420$^{\circ}$C. The latter transition was reversible and repeatable. While the transition temperatures are shifted relative to the values known for bulk GeTe, the nanoparticle-based samples still exhibit the same structural phases reported for sputtered GeTe. Thus, colloidal GeTe maintains the same general phase behavior as bulk GeTe while allowing for more flexible and accessible fabrication. Therefore, nanoparticle-based GeTe films show great potential for applications, such as in active photonics.

cond-mat.mtrl-sci↗
arXiv2019

The Gauss-Bonnet formula of a conical metric on a compact Riemann surface

We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric. We also construct explicitly some conical metrics whose curvature is not integrable.

math.DG↗