Guan/Li Inverse-Type Flow & Geometric Inequalities

• Guan/Li inverse-type flow is a fully nonlinear evolution method that preserves quermassintegrals and aids in establishing sharp geometric inequalities.
• It employs locally constrained normal velocities based on inverse curvature functions to maintain convexity and horo-convexity in diverse geometries.
• Rigorous a priori estimates for curvature and support functions ensure smooth long-time convergence to geodesic spheres and validate the derived inequalities.

The Guan/Li flow of inverse type is a class of fully nonlinear geometric evolution equations for hypersurfaces, designed to preserve or monotonically evolve certain geometric integrals, especially quermassintegrals, and to establish fundamental geometric inequalities in both spherical and hyperbolic geometries. These flows generalize inverse curvature flows and are characterized by locally constrained normal velocities of inverse curvature type, tailored to maintain convexity under geometric constraints defined by support or curvature functions.

1. Background and Context

The Guan/Li inverse-type flow originated in the study of curvature evolution equations preserving geometric invariants, particularly in the context of proving geometric inequalities such as the Alexandrov–Fenchel and Minkowski-type inequalities. The flows originally emerged in the Euclidean and hyperbolic settings and have been generalized to the sphere to tackle the quermassintegral inequalities for new convexity classes such as horo-convexity.

In the sphere, horo-convexity is defined as a strengthening of convexity for subsets in the unit sphere, analogous to horo-convexity in hyperbolic space. Pan and Scheuer established the suitability of the Guan/Li flow of inverse type for horo-convex domains, proving smooth long-time convergence and using these results to derive a full suite of quermassintegral inequalities (Pan et al., 14 Dec 2025).

Extensions of the flow in the hyperbolic setting, particularly the Brendle–Guan–Li (BGL) flow, have been crucial in proving sharp Michael–Simon and Alexandrov–Fenchel inequalities, as shown in work by Cui and Zhao (Cui et al., 2022). More general formulations and associated convergence results are encompassed in the framework detailed by Ding and Li (Ding et al., 2023).

2. Definition of the Guan/Li Inverse-Type Flow

The classical formulation for the Guan/Li inverse-type flow on a hypersurface $M$ in the northern hemisphere $S^{n+1}_+$ of the sphere or in hyperbolic space is as follows. Fix a symmetric curvature function $F(\kappa_1, ..., \kappa_n)$ defined on the positive cone $\Gamma_+ = \{\kappa_i > 0\}$, which is homogeneous of degree 1, strictly increasing in each argument, concave, and inverse-concave.

In the spherical setting, given the notations

• $r(p) = d_{S^{n+1}}(p,N)$, distance to a fixed pole $N$
• $\phi(r) = \sin r$, $\phi'(r) = \cos r$
• Support function $u(p) = \langle \partial_r, \nu(p)\rangle \phi'(r)$,

the locally constrained Guan/Li flow reads: $$\partial_t x(p, t) = \left( \frac{ \phi'(r) }{F(\kappa(p, t))} - u(p, t) \right) \nu(p, t), \qquad x(\cdot,0) = M_0$$ A central special case uses the quotient of elementary symmetric polynomials: $$F = \frac{H_k}{H_{k-1}}, \quad H_j = \frac1{ {n \choose j} } \sum_{1 \leq i_1 < \dots < i_j \leq n} \kappa_{i_1} \cdots \kappa_{i_j}$$ leading to a flow which preserves the $k$th quermassintegral $W_k$ and strictly increases $W_{k-1}$ (Pan et al., 14 Dec 2025).

In hyperbolic space $\mathbb{H}^{n+1}$, with $\lambda(r) = \sinh r$ and $\lambda'(r) = \cosh r$, analogous flows take the form: $$\partial_t X = \left( \frac{\lambda'(r)}{H} - u \right) \nu$$ or, more generally, for higher mean curvatures with $F = \lambda'(r) E_{k-1} / E_k - u$, where $E_k$ are normalized symmetric polynomials in the principal curvatures (Cui et al., 2022).

3. Horo-Convexity and Geometric Constraints

A $C^2$-hypersurface $M \subset S^{n+1}_+$ is horo-convex if

$\phi'(r) h_{ij} \geq (1-u) g_{ij}$

or, equivalently, the tensor $S^i_j = \phi' h^i_j + (u-1) \delta^i_j$ is nonnegative definite. This geometric constraint generalizes strict convexity and is critical for ensuring preservation of convexity and avoidance of degeneracies under the flow. The tensor $S^i_j$ satisfies a fully nonlinear parabolic inequality, implying that horo-convexity is preserved by the viscocity-max-eigenvalue lemma throughout the evolution (Pan et al., 14 Dec 2025).

Similarly, in the hyperbolic setting, the initial hypersurface is required to be star-shaped and $h$-convex (all shifted principal curvatures positive), and the weight functions involved in monotonicity arguments must satisfy specific radial monotonicity or ODE constraints (Cui et al., 2022).

4. Analytical Framework: Evolution Equations and A Priori Estimates

The analysis of the flow centers on deriving sharp a priori estimates and elucidating the evolution of geometric quantities:

• Distance-to-pole (or origin) estimate: The evolution of the support function or $\phi'(r)$ ensures that hypersurfaces remain strictly inside the domain of definition (hemisphere or hyperbolic ball), and evolve away from boundary degeneracy.
• Lower and upper curvature bounds: The horo-convexity condition gives a time-uniform lower bound for all principal curvatures, while time-dependent and then time-independent upper bounds are achieved using the maximum principle applied to logarithmic quantities involving the largest principal curvature and the support function (e.g., $w = \ln \kappa_{\max} - \ln u$).
• Preservation of convexity: Further use of maximum principles and tensor parabolic principles ensures convexity or horo-convexity is maintained for the duration of the flow.
• $C^2$- and higher-order estimates: Uniform bounds on second fundamental form, area elements, and derivatives up to arbitrary order follow by standard parabolic regularity theory (Krylov–Safonov/Schauder).

Analogous estimates in the Euclidean setting are achieved by barrier arguments with spheres and explicit test functions involving support and radial functions (Ding et al., 2023).

5. Long-Time Existence and Convergence

Uniform $C^2$ bounds and regularity ensure global existence in time ($T^* = \infty$). Using compactness arguments and regularity theory, hypersurfaces evolving by the Guan/Li flow are shown to converge smoothly (in $C^\infty$) to a standard geodesic sphere (in the spherical or hyperbolic sense), centered at the respective pole. This convergence is deduced from the constancy of the support function and application of the Arzelà–Ascoli theorem together with identification of limit flows via the maximum principle (Pan et al., 14 Dec 2025, Ding et al., 2023).

6. Applications to Geometric Inequalities

The primary application of the Guan/Li inverse-type flow is the proof of sharp geometric inequalities for convex or horo-convex hypersurfaces:

• Spherical Quermassintegral Inequalities: For horo-convex hypersurfaces in $S^{n+1}_+$, the flow and the preserved quantities yield

$W_k(M) \geq f_k(W_{k-1}(M)), \quad k=1,\ldots, n$

with equality precisely for geodesic spheres. These establish the full family of quermassintegral inequalities in the spherical setting, generalizing earlier hyperbolic results (Pan et al., 14 Dec 2025).

• Hyperbolic Michael–Simon and Minkowski-type Inequalities: In $\mathbb{H}^{n+1}$, using the Brendle–Guan–Li inverse flow, sharp weighted and unweighted inequalities for mean and higher curvatures are established, including cases corresponding to previously unresolved Alexandrov–Fenchel inequalities (Cui et al., 2022). Monotonicity of appropriately constructed functionals under the flow is central to these arguments.

7. Generalizations and Frameworks

The Ding–Li framework (Ding et al., 2023) generalizes the Guan/Li flow as a class of anisotropic inverse-curvature–type flows, formulated for hypersurfaces in $\mathbb{R}^{n+1}$ (and adaptable to other ambient geometries) with flows of the form

$$\partial_t X = Q(X, \nu, \kappa) \nu, \quad Q = G(X, \nu) F(\kappa)^{-\beta}$$

Here, $F$ is a smooth, elliptic, degree-1 homogeneous curvature function, and $G$ is a positive function ensuring admissibility and necessary monotonicity properties. For suitable choice (e.g., $F = \sigma_k^{1/k}$, $$G = \psi(\frac{X}{|X|})(X \cdot \nu)^{1/k} |X|^{-(n+1)/k}$$), the Guan/Li flow is recovered as a special case.

Under log-convexity conditions on $uG$ and star-shaped, $k$-convex initial data, the flow yields smooth solutions for all time and convergence (after suitable rescaling) to unique solutions of corresponding elliptic prescribed curvature equations.

Main Contribution	Authors/Source	arXiv ID
Spherical quermassintegral inequalities, horo-convexity, and smooth convergence of Guan/Li flow	Pan & Scheuer	(Pan et al., 14 Dec 2025)
Hyperbolic Michael–Simon and Alexandrov–Fenchel inequalities via Brendle–Guan–Li flow	Cui & Zhao	(Cui et al., 2022)
General curvature flow framework (inc. Guan–Li) and existence/convergence theorems	Ding & Li	(Ding et al., 2023)