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arxiv: 2605.01979 · v1 · submitted 2026-05-03 · 🧮 math.AP

Sets of finite perimeter on Riemannian manifolds and stochastic completeness

Luca Gennaioli This is my paper

Pith reviewed 2026-05-09 16:09 UTC · model grok-4.3

classification 🧮 math.AP
keywords sets of finite perimetertotal variationheat semigroupBV functionsweighted Riemannian manifoldsstochastic completeness
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The pith

The total variation of compactly supported BV functions equals a limit involving the heat semigroup on any smooth complete weighted Riemannian manifold.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the total variation of a compactly supported function of bounded variation on a weighted Riemannian manifold can be recovered exactly as the limit as time goes to zero of an expression built from the heat semigroup acting on the function. This equivalence extends an earlier result that applied only in more restricted settings. The authors also exhibit a weighted manifold on which the same limit expression fails to recover the total variation for a broad family of sets of finite perimeter.

Core claim

We prove a heat semigroup characterization of the total variation for compactly supported BV on arbitrary smooth complete weighted Riemannian manifolds, extending the main result in GP15. We then provide an example of a weighted manifold where such equivalence does not hold for a large class of sets of finite perimeter.

What carries the argument

The heat semigroup on the weighted manifold, used to express total variation as the limit of a normalized difference between a function and its evolved version under the semigroup.

If this is right

  • The characterization applies uniformly to all smooth complete weighted manifolds without further curvature or volume assumptions.
  • Sets of finite perimeter inherit the same semigroup representation of their perimeter measure.
  • The result is false on manifolds that lack the necessary semigroup properties, as shown by the explicit counterexample.
  • Stochastic completeness of the manifold is tied to whether the characterization holds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The counterexample indicates that stochastic completeness is the precise geometric condition separating manifolds where the characterization succeeds from those where it fails.
  • The approach may extend to other notions of variation or to non-compactly supported functions once the semigroup properties are verified.
  • It connects the analytic theory of BV functions directly to the long-time behavior of the heat flow on the manifold.

Load-bearing premise

The heat semigroup must exist on the weighted manifold and satisfy the regularity and contraction properties needed to make the limit expression match total variation.

What would settle it

A concrete weighted manifold on which the heat semigroup fails to satisfy the required bounds, so that the semigroup-based limit differs from the total variation for some compactly supported BV functions.

read the original abstract

We prove a heat semigroup characterization of the total variation for compactly supported ${\rm BV}$ on arbitrary smooth complete weighted Riemannian manifolds, extending the main result in \cite{GP15}. We then provide an example of a weighted manifold where such equivalence does not hold for a large class of sets of finite perimeter.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proves a heat semigroup characterization of the total variation for compactly supported BV functions on arbitrary smooth complete weighted Riemannian manifolds, extending the main result of GP15. It then constructs a counterexample weighted manifold on which the equivalence fails for a large class of sets of finite perimeter.

Significance. The counterexample construction is a clear strength, as it demonstrates the necessity of stochastic completeness for the characterization to hold and provides a concrete falsifiable test case. If the main theorem is correctly stated under this hypothesis, the work would usefully extend the GP15 result to the weighted setting while clarifying the role of mass conservation in the heat-kernel approximation of perimeter.

major comments (2)
  1. [Abstract] Abstract: the claim that the characterization holds for BV on 'arbitrary smooth complete weighted Riemannian manifolds' is directly contradicted by the counterexample constructed later in the paper. The proof of the positive result must therefore rely on stochastic completeness (ensuring the heat semigroup is conservative), which is not automatic for every complete weighted manifold; this assumption must be stated explicitly in the main theorem.
  2. [Main theorem / proof of characterization] The passage from the heat-kernel representation to the total-variation formula (the load-bearing identification of the perimeter measure) is expected to break without mass conservation. The counterexample should be used to isolate the precise step where the approximation arguments fail, rather than leaving the scope of the positive result ambiguous.
minor comments (1)
  1. [Title and Abstract] The title emphasizes stochastic completeness, yet the abstract does not; align the two so that the dependence on this condition is visible at first reading.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and helpful report. The comments correctly identify an inconsistency in the presentation of the main result and suggest a useful way to strengthen the discussion of the counterexample. We will address both points in a revised version of the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the characterization holds for BV on 'arbitrary smooth complete weighted Riemannian manifolds' is directly contradicted by the counterexample constructed later in the paper. The proof of the positive result must therefore rely on stochastic completeness (ensuring the heat semigroup is conservative), which is not automatic for every complete weighted manifold; this assumption must be stated explicitly in the main theorem.

    Authors: We agree with this observation. The positive characterization result requires the weighted manifold to be stochastically complete, which guarantees that the heat semigroup preserves mass. Although this is part of the context in extending GP15 and is reflected in the paper's title, it was not explicitly stated in the abstract or theorem. We will revise the abstract to read 'stochastically complete smooth complete weighted Riemannian manifolds' and add the assumption to the statement of the main theorem. The counterexample is constructed on a manifold that fails stochastic completeness. revision: yes

  2. Referee: [Main theorem / proof of characterization] The passage from the heat-kernel representation to the total-variation formula (the load-bearing identification of the perimeter measure) is expected to break without mass conservation. The counterexample should be used to isolate the precise step where the approximation arguments fail, rather than leaving the scope of the positive result ambiguous.

    Authors: We concur that clarifying this point strengthens the paper. In the revised manuscript, we will expand the discussion following the counterexample to identify the specific step in the proof where mass conservation is used: namely, in justifying the limit passage for the total variation approximation via the heat kernel, where the integral of the heat kernel equals 1 only under stochastic completeness. Without it, the inequality becomes strict, breaking the equality. This will make the scope unambiguous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends external prior work independently

full rationale

The paper states a heat-semigroup characterization of total variation for compactly supported BV functions, explicitly extending the main result of the external reference GP15, and separately supplies a counterexample manifold where the equivalence fails. No equations, definitions, or steps in the abstract or described structure reduce the claimed result to a fitted parameter, self-referential quantity, or self-citation chain by construction. The central proof relies on standard analysis techniques and the cited prior theorem rather than redefining inputs as outputs, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the proof presumably relies on standard existence and properties of the heat semigroup on complete Riemannian manifolds, which are drawn from prior literature rather than introduced here.

axioms (1)
  • standard math Existence and basic contraction properties of the heat semigroup on smooth complete weighted Riemannian manifolds
    Invoked implicitly to define the characterization; standard background in the field.

pith-pipeline@v0.9.0 · 5324 in / 1110 out tokens · 31169 ms · 2026-05-09T16:09:06.748820+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 9 canonical work pages

  1. [1]

    ---to appear--- , year =

    Caselli, Michele and Florit Simon, Enric and Serra, Joaquim , title =. ---to appear--- , year =

  2. [2]

    ---preprint--- , year =

    Caselli, Michele and Gennaioli, Luca , title =. ---preprint--- , year =

  3. [3]

    Fractional Calculus and Applied Analysis , year =

    Carbotti, Alessandro and Cito, Simone and La Manna, Domenico Angelo and Pallara, Diego , title =. Fractional Calculus and Applied Analysis , year =. doi:https://doi.org/10.1007/s13540-022-00066-8 , URL =

    work page doi:10.1007/s13540-022-00066-8

  4. [4]

    Grigor'yan, Alexander , TITLE =. Bull. Amer. Math. Soc. (N.S.) , FJOURNAL =. 1999 , NUMBER =. doi:10.1090/S0273-0979-99-00776-4 , URL =

    work page doi:10.1090/s0273-0979-99-00776-4 1999

  5. [5]

    Free boundary minimal surfaces with connected boundary and arbi- trary genus

    Federer, Herbert and Fleming, Wendell H. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1960 , PAGES =. doi:10.2307/1970227 , URL =

    work page doi:10.2307/1970227 1960

  6. [6]

    Caselli, Michele and Florit-Simon, Enric and Serra, Joaquim , TITLE =. Math. Ann. , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s00208-024-02894-w , URL =

    work page doi:10.1007/s00208-024-02894-w 2024

  7. [7]

    Heat semigroup and functions of bounded variation on Riemannian manifolds

    Miranda, Jr., M. and Pallara, D. and Paronetto, F. and Preunkert, M. , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2007 , PAGES =. doi:10.1515/CRELLE.2007.093 , URL =

    work page doi:10.1515/crelle.2007.093 2007

  8. [8]

    Grigor’yan, Heat Kernel and Analysis on Manifolds, volume 47 ofAMS/IP Studies in Advanced Mathematics, AMS, 2009

    Grigor'yan, Alexander , TITLE =. 2009 , PAGES =. doi:10.1090/amsip/047 , URL =

    work page doi:10.1090/amsip/047 2009

  9. [9]

    Atti Accad

    De Giorgi, Ennio , TITLE =. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) , FJOURNAL =. 1953 , PAGES =

    1953

  10. [10]

    De Giorgi, Ennio , TITLE =. Ann. Mat. Pura Appl. (4) , FJOURNAL =. 1954 , PAGES =. doi:10.1007/BF02412838 , URL =

    work page doi:10.1007/bf02412838 1954

  11. [11]

    Ricerche Mat

    De Giorgi, Ennio , TITLE =. Ricerche Mat. , FJOURNAL =. 1955 , PAGES =

    1955

  12. [12]

    G\"uneysu, Batu and Pallara, Diego , TITLE =. Math. Ann. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00208-015-1208-x , URL =

    work page doi:10.1007/s00208-015-1208-x 2015

  13. [13]

    and Mauceri, G

    Carbonaro, A. and Mauceri, G. , TITLE =. Bull. Austral. Math. Soc. , FJOURNAL =. 2007 , NUMBER =. doi:10.1017/S000497270003954X , URL =

    work page doi:10.1017/s000497270003954x 2007