From the discrete to the continuous, from simplicial complexes to Riemannian manifolds. Approximating flows and cuts on manifolds by discrete versions (2512.05319v1)

Abstract: Many fundamental structures of Riemannian geometry have found discrete counterparts for graphs or combinatorial ones for simplicial complexes. These include those discussed in this survey, Hodge theory, Morse theory, the spectral theory of Laplace type operators and Cheeger inequalities, and their interconnections. This raises the question of the relation between them, abstractly as structural analogies and concretely what happens when a graph constructed from random sampling of a Riemannian manifold or a simplicial complex triangulating such a manifold converge to that manifold. We survey the current state of research, highlighting some recent developments like Cheeger type inequalities for the higher dimensional geometry of simplicial complexes, Floer type constructions in the presence of periodic or homoclinic orbits of dynamical systems or the disorientability of simplicial complexes.

• The paper bridges discrete and continuous geometric structures by matching Laplacians, Morse theory, and Cheeger inequalities.
• It employs Lovász extensions and variational convergence methods to rigorously link simplicial complexes with Riemannian manifolds.
• The work provides actionable insights for data analysis and machine learning through spectral clustering and manifold approximation techniques.

Discrete-to-Continuous Analogies: Laplacians, Morse Theory, and Cheeger Inequalities Across Simplicial Complexes and Riemannian Manifolds

Introduction

The paper presents a comprehensive and technically rigorous synthesis of structural analogies and convergence relations between discrete, combinatorial, and smooth geometric objects: graphs, simplicial complexes, and Riemannian manifolds. Focusing on Laplace-type operators (combinatorial/discrete and continuous), the authors systematically build connections among Hodge/Eckmann theories, Morse-Witten-Floer/Forman homology, and Cheeger-type isoperimetric inequalities. This approach clarifies both direct analogs and limiting behavior, illuminating how discrete objects approximate their geometric counterparts and, reciprocally, how discrete insights enhance geometric analysis.

Discrete and Continuous Laplacians

In the continuous setting, the Laplace-Beltrami operator provides deep geometric information about Riemannian manifolds, with its spectrum encoding isoperimetric and topological properties. Its generalization to differential forms—the Hodge Laplacian—furnishes a bridge to cohomology via the celebrated Hodge theorem. On the combinatorial side, Eckmann’s Laplacian for simplicial complexes follows—via boundary and coboundary operators, their adjoints, and appropriate choice of scalar product—from the structure of the chain and cochain complexes. The spectrum of the discrete Laplacian reveals topological invariants such as Betti numbers and, by extension, geometric features when the complex triangulates a space with a metric.

A crucial technical insight is that, for both settings, the up and down Laplacians and the full Laplacian are positive semi-definite, self-adjoint, and admit Hodge decompositions, with the kernel encoding cohomology: the multiplicity of the zero eigenvalue yields the Betti numbers. The nonzero spectrum captures more subtle geometric and combinatorial structure.

Rayleigh Quotients and $p$-Laplacians

The paper carefully extends the classical Rayleigh quotient framework, showing all eigenvalues of symmetric Laplacians (continuous or discrete) correspond to critical values of associated energy functionals. The transition to $p$-Laplacians, with emphasis on the $p=1$ (“1-Laplacian”) case, captures nonlinear diffusion and is directly tied to vertex or simplex isoperimetry, including the Cheeger constant.

The authors underline the subtlety that, for nonlinear Laplacians, the count of eigenvalues can exceed the space’s dimension and that minimax-type eigenvalues (via the Krasnoselskii genus) refine Cheeger’s estimates. Monotonicity with respect to $p$ yields increasingly tight control of isoperimetric constants as $p\to 1$.

Lovász Extensions and Variational Convergence

The Lovász extension is presented as a crucial tool to pass from set functions on combinatorial objects (e.g., submodular cut functions on graphs or complexes) to continuous convex functionals on real vector spaces. This enables precise formulations of variational convergence (e.g., $\Gamma$-convergence of functionals, convergence of eigenstructures), supporting strong connections between discrete approximations (triangulations, sampled graphs) and their limiting Riemannian objects.

Morse Theory: Smooth, Combinatorial, and Floer Extensions

The exposition presents a unified account of Morse theory in both smooth and discrete contexts. In the smooth setting, Morse inequalities relate counts of critical points (enumerated by index) of smooth functions to Betti numbers, with the Witten Laplacian providing a spectral mechanism to recover these inequalities analytically. The localization of low-lying eigenfunctions of the Witten Laplacian near critical points, especially in the semiclassical limit, is explicitly connected to the topology of the underlying manifold.

Discrete Morse theory is introduced via Forman’s framework, wherein critical simplices are characterized by discrete analogs of non-degeneracy and the assignment of appropriate arrow structures. The deep result here is an equivalence theorem: critical points of a discrete Morse function correspond precisely, via the Lovász extension, to metric and PL critical points of the extended function on the geometric realization of the barycentric subdivision. This provides a formal mechanism to translate combinatorial Morse data into the PL or metric category.

Extensions to Morse–Smale flows, periodic and homoclinic orbits (via Franks perturbation and Floer-type boundary operators), reveal how the boundary operator—constructed from counting oriented gradient flowlines—remains algebraically robust even in the presence of more complex dynamics. The combinatorial analog uses zigzag flows on simplices, and the paper carefully describes the algebraic structures involved.

Cheeger-Type Inequalities: Graphs, Manifolds, and Higher Dimensions

The central quantitative link between spectral data and isoperimetric structure is the Cheeger inequality. For Riemannian manifolds, the Cheeger constant quantifies the minimal boundary-to-volume ratio cut, and Cheeger/Buser’s estimates envelop the spectral gap $\lambda_2$ of the Laplacian. For (weighted or signed) graphs, analogous reasoning yields tight bounds, with precise constants depending on normalization conventions and multiplicities.

The extension to higher-dimensional simplicial complexes is technical and achieves a main contribution of the survey. By associating signed graphs to $k$-simplices (whose adjacency is encoded by co-faceting in $k+1$ dimension), the authors identify conditions under which the spectrum of the up-Laplacian can be precisely controlled by higher-order Cheeger constants. Multiple, ostensibly distinct, definitions of the combinatorial Cheeger constant are developed and rigorously proved equivalent, encompassing formulations via multiplicities, integer-valued expanders, 1-Laplacian eigenvalues, and filling radius.

For the largest eigenvalue of the up-Laplacian, an ‘anti-Cheeger’ duality is established (using signed bipartiteness ratios), tight upper and lower bounds are provided, and structural obstructions (disorientability, existence of twisted cycles, and branching) are analyzed. Theorems characterizing when these eigenvalues attain their extrema in terms of complex orientation properties and the minimal cycles needed for disorientability have important implications for spectral topology.

Convergence and Data Analysis Implications

Sampling-based graph approximations to Riemannian manifolds under appropriate rescaling of the graph Laplacian’s spectrum are shown to converge to their smooth counterparts. Convergence of the perimeter functionals (via $\Gamma$-convergence) and the stability of Cheeger constants under this approximation are established, with direct impact on the analysis of data-driven Laplacians (e.g., manifold learning, spectral clustering).

The survey identifies challenges for higher-order objects: while random walks on graphs (and their relation to the Laplacian) exhibit strong ergodic properties, extensions to random walks or Markov processes on higher-dimensional complexes lack full analogs, with open questions regarding spectra and limiting distributions.

Conclusion

This paper integrates the paper of geometric, combinatorial, and analytic structures by establishing deep analogies and rigorous convergence results among Laplacians, Morse and Floer theories, and Cheeger-type inequalities across manifolds, graphs, and simplicial complexes. The explicit bridging of the discrete-to-continuous gap, both structurally and quantitatively, is achieved via sophisticated use of extensions (Lovász, variational principles), combinatorial–geometric constructions (order complexes, barycentric subdivisions), and spectral theory.

Strong claims include the precise equivalence of disparate formulations of higher-dimensional Cheeger constants and the match between discrete and continuous Morse theory critical structures via the Lovász extension. The survey also supplies a systematic framework for understanding when and how discrete analogs recover or inform continuous geometric invariants.

The practical implications for data science and machine learning are significant: these theories underpin the foundations of graph- and complex-based learning algorithms, graph clustering, and geometric inference.

Several open directions are identified: understanding the geometric role of higher-order Laplacian spectra, constructing well-behaved Markov processes on simplicial complexes, and extending Morse–Floer-type invariants in discrete settings. These advances are expected to reshape both the mathematical landscape of geometric analysis and the methodological toolkit for high-dimensional data.

All references in the essay correspond to the internal references and citations within the paper "From the discrete to the continuous, from simplicial complexes to Riemannian manifolds. Approximating flows and cuts on manifolds by discrete versions" (2512.05319).