A comparison theorem with applications to sharp geometric inequalities for submanifolds

Abstract: Inspired by the work of Cordero-Erausquin, McCann and Schmuckenschläger [{\it Invent. Math.,} 2001], we derive an explicit expression for the Jacobian determinant of the normal exponential map on a submanifold, establishing a relationship with its ambient counterpart. This formula leads to a new comparison theorem which is closely related to the comparison theorem of Heintze-Karcher [{\it Ann. Sci. École Norm. Sup.,} 1978] and the esitimate of Brendle [{\it Comm. Pure Appl. Math.,} 2023]. As applications, inspired by Wang {\it Ann. Fac. Sci. Toulouse Math.,} 2023, we obtain a Fenchel-Borsuk-Chern-Lashof-type inequality and a Willmore-Chen-type inequality on closed submanifolds in complete noncompact manifolds with nonnegative curvature and Euclidean volume growth.

• The paper develops an explicit Jacobian formula and comparison theorem for normal exponential maps, including an n-Ricci curvature version that extends higher-codimension Heintze–Karcher methods.
• The resulting sharp inequalities bound total absolute curvature and total mean curvature by AVR(M) times the corresponding sphere constants, with detailed rigidity conditions characterizing equality.
• The Willmore–Chen inequality rules out closed minimal submanifolds in intermediate dimensions under nonnegative n-Ricci curvature and Euclidean volume growth, while highlighting open questions about weaker hypotheses and ambient rigidity.

Background and motivation

The paper by Pan and Yi (2605.06074) develops a comparison theory for the normal exponential map of submanifolds and applies it to two classical families of sharp integral inequalities: the Fenchel–Borsuk–Chern–Lashof inequality for total absolute curvature and the Willmore–Chen inequality for total mean curvature. The setting is a complete noncompact ambient manifold $(M^{n+m},\bar g)$ with Euclidean volume growth, i.e. $\mathrm{AVR}(M,\bar g)>0$, where

$$\mathrm{AVR}(M,\bar g):=\lim_{r\to\infty}\frac{|B_r(p)|}{\omega_{n+m}r^{n+m}}.$$

The Euclidean prototypes are Chern–Lashof's bound $$\int_\Sigma K^*(x)\,d\mathrm{vol}_\Sigma\ge 2|\mathbb{S}^{n+m-1}|$$ with equality exactly for convex hypersurfaces in a linear subspace, and Chen's generalization of Willmore's inequality, $$\int_\Sigma|\mathbf H|^n\,d\mathrm{vol}_\Sigma\ge |\mathbb{S}^n|$$. In the Riemannian setting, Agostiniani–Fogagnolo–Mazzieri proved a Willmore–Chen type inequality for hypersurfaces under nonnegative Ricci curvature, with constant $\mathrm{AVR}(M,\bar g)$; Wang later gave a comparison-theoretic proof via the area formula for the normal exponential map. The present paper extends this program to higher codimension.

A key structural observation motivates the choice of curvature hypothesis: the Eguchi–Hanson metric on $T\mathbb{S}^2$ is Ricci flat with Euclidean volume growth yet contains a totally geodesic closed $\mathbb{S}^2$, so nonnegative Ricci curvature alone cannot yield a codimension-free Willmore–Chen inequality. The authors argue that the correct intermediate condition is Bishop's $k$-Ricci curvature,

$$\mathrm{Ric}^{M}_{k}(v,P):=\sum_{i=1}^{k}\bar R(v,e_i,e_i,v),$$

which interpolates between sectional ($\mathrm{AVR}(M,\bar g)>0$0) and Ricci ($\mathrm{AVR}(M,\bar g)>0$1) curvature and is naturally adapted to $\mathrm{AVR}(M,\bar g)>0$2-dimensional submanifolds.

Jacobian formula for the normal exponential map

The central technical device is an explicit decomposition of the differential of the normal exponential map $\mathrm{AVR}(M,\bar g)>0$3. Adopting a trick of Cordero-Erausquin–McCann–Schmuckenschläger, the authors factor the map as $\mathrm{AVR}(M,\bar g)>0$4 where $\mathrm{AVR}(M,\bar g)>0$5 on the pullback bundle $\mathrm{AVR}(M,\bar g)>0$6 (the gradient field term is included for generality but not used subsequently). Along a geodesic $\mathrm{AVR}(M,\bar g)>0$7 avoiding cut points, they compute the Jacobi fields generated by variations of base point and fiber coordinate and obtain, in adapted frames, a block-triangular matrix identity whose frame-independent consequence is

$\mathrm{AVR}(M,\bar g)>0$8

where $\mathrm{AVR}(M,\bar g)>0$9 is the restriction to $$\mathrm{AVR}(M,\bar g):=\lim_{r\to\infty}\frac{|B_r(p)|}{\omega_{n+m}r^{n+m}}.$$0 of $$\mathrm{AVR}(M,\bar g):=\lim_{r\to\infty}\frac{|B_r(p)|}{\omega_{n+m}r^{n+m}}.$$1. This extends classical formulas of do Carmo and Brendle from Euclidean space to arbitrary ambient manifolds, and it relates directly to Brendle's Alexandrov–Bakelman–Pucci-based estimate used in his work on Sobolev and isoperimetric inequalities.

Two corollaries follow immediately. First, focal points along $$\mathrm{AVR}(M,\bar g):=\lim_{r\to\infty}\frac{|B_r(p)|}{\omega_{n+m}r^{n+m}}.$$2 occur precisely when the tensor $$\mathrm{AVR}(M,\bar g):=\lim_{r\to\infty}\frac{|B_r(p)|}{\omega_{n+m}r^{n+m}}.$$3 degenerates (assuming infinite injectivity radius). Second, the authors introduce a localized cut-distance function

$$\mathrm{AVR}(M,\bar g):=\lim_{r\to\infty}\frac{|B_r(p)|}{\omega_{n+m}r^{n+m}}.$$4

to handle immersions that are not embeddings, for which $$\mathrm{AVR}(M,\bar g):=\lim_{r\to\infty}\frac{|B_r(p)|}{\omega_{n+m}r^{n+m}}.$$5 may be discontinuous or vanish. They prove $$\mathrm{AVR}(M,\bar g):=\lim_{r\to\infty}\frac{|B_r(p)|}{\omega_{n+m}r^{n+m}}.$$6, so it inherits continuity where finite — a pointwise identity connecting cut points of the immersion, focal points, and ambient cut points that underlies all subsequent rigidity arguments.

A monotonicity result and the new comparison theorem

As a byproduct, before proving the main comparison theorem, the authors establish a monotonicity property in the Hessian/Laplacian comparison theorem that appears to be new. If $$\mathrm{AVR}(M,\bar g):=\lim_{r\to\infty}\frac{|B_r(p)|}{\omega_{n+m}r^{n+m}}.$$7 along a minimal geodesic $$\mathrm{AVR}(M,\bar g):=\lim_{r\to\infty}\frac{|B_r(p)|}{\omega_{n+m}r^{n+m}}.$$8 and $$\mathrm{AVR}(M,\bar g):=\lim_{r\to\infty}\frac{|B_r(p)|}{\omega_{n+m}r^{n+m}}.$$9 are orthonormal, then differentiating the index-form representation of $$\int_\Sigma K^*(x)\,d\mathrm{vol}_\Sigma\ge 2|\mathbb{S}^{n+m-1}|$$0 with respect to $$\int_\Sigma K^*(x)\,d\mathrm{vol}_\Sigma\ge 2|\mathbb{S}^{n+m-1}|$$1 yields

$$\int_\Sigma K^*(x)\,d\mathrm{vol}_\Sigma\ge 2|\mathbb{S}^{n+m-1}|$$2

and combining this with Bishop's Jacobi-field wedge estimate gives both a derivative bound below the model rate and a ratio monotonicity:

$$\int_\Sigma K^*(x)\,d\mathrm{vol}_\Sigma\ge 2|\mathbb{S}^{n+m-1}|$$3

The dual statement holds under an upper sectional curvature bound. The proof hinges on the smooth dependence of the Jacobi fields $$\int_\Sigma K^*(x)\,d\mathrm{vol}_\Sigma\ge 2|\mathbb{S}^{n+m-1}|$$4 (with $$\int_\Sigma K^*(x)\,d\mathrm{vol}_\Sigma\ge 2|\mathbb{S}^{n+m-1}|$$5, $$\int_\Sigma K^*(x)\,d\mathrm{vol}_\Sigma\ge 2|\mathbb{S}^{n+m-1}|$$6) on the parameter $$\int_\Sigma K^*(x)\,d\mathrm{vol}_\Sigma\ge 2|\mathbb{S}^{n+m-1}|$$7, established via an explicit basis-solution construction.

The main comparison theorem combines this with Lemma 4 to compare $$\int_\Sigma K^*(x)\,d\mathrm{vol}_\Sigma\ge 2|\mathbb{S}^{n+m-1}|$$8 against its model counterpart in the space form $$\int_\Sigma K^*(x)\,d\mathrm{vol}_\Sigma\ge 2|\mathbb{S}^{n+m-1}|$$9. Two regimes are treated:

• Sectional curvature regime: if $$\int_\Sigma|\mathbf H|^n\,d\mathrm{vol}_\Sigma\ge |\mathbb{S}^n|$$0 and $$\int_\Sigma|\mathbf H|^n\,d\mathrm{vol}_\Sigma\ge |\mathbb{S}^n|$$1, then

$$\int_\Sigma|\mathbf H|^n\,d\mathrm{vol}_\Sigma\ge |\mathbb{S}^n|$$2

with ratio monotonicity in $$\int_\Sigma|\mathbf H|^n\,d\mathrm{vol}_\Sigma\ge |\mathbb{S}^n|$$3 and equality rigidifying the ambient metric to constant curvature $$\int_\Sigma|\mathbf H|^n\,d\mathrm{vol}_\Sigma\ge |\mathbb{S}^n|$$4 on all tangent planes containing $$\int_\Sigma|\mathbf H|^n\,d\mathrm{vol}_\Sigma\ge |\mathbb{S}^n|$$5.

• $$\int_\Sigma|\mathbf H|^n\,d\mathrm{vol}_\Sigma\ge |\mathbb{S}^n|$$6-Ricci regime: if $$\int_\Sigma|\mathbf H|^n\,d\mathrm{vol}_\Sigma\ge |\mathbb{S}^n|$$7 and the model submanifold is umbilical with $$\int_\Sigma|\mathbf H|^n\,d\mathrm{vol}_\Sigma\ge |\mathbb{S}^n|$$8, then

$$\int_\Sigma|\mathbf H|^n\,d\mathrm{vol}_\Sigma\ge |\mathbb{S}^n|$$9

again with rigidity.

The second assertion weakens the lower sectional curvature hypothesis of Heintze–Karcher to an $\mathrm{AVR}(M,\bar g)$0-Ricci lower bound in higher codimension — the authors state plainly that they were unable to achieve this weakening within the original Heintze–Karcher framework and had to introduce the new approach. They note further consequences obtainable by splicing their theorem into the Heintze–Karcher argument, but do not carry these out. The upper-curvature side of Heintze–Karcher is left untouched.

Fenchel–Borsuk–Chern–Lashof inequality

Theorem (Chern–Lashof type). Let $\mathrm{AVR}(M,\bar g)$1 be complete, noncompact, with $\mathrm{AVR}(M,\bar g)$2 and Euclidean volume growth, and let $\mathrm{AVR}(M,\bar g)$3 be an isometric immersion of a closed manifold. Then

$\mathrm{AVR}(M,\bar g)$4

Equality holds if and only if (i) at each elliptic point $\mathrm{AVR}(M,\bar g)$5 the image of $\mathrm{AVR}(M,\bar g)$6 is one-dimensional; (ii) $\mathrm{AVR}(M,\bar g)$7 is a diffeomorphism onto its image with pulled-back metric $\mathrm{AVR}(M,\bar g)$8; and (iii) off $\mathrm{AVR}(M,\bar g)$9 every shape operator $T\mathbb{S}^2$0 has a zero eigenvalue. Equality additionally forces $T\mathbb{S}^2$1 connected and $T\mathbb{S}^2$2 an embedding.

The proof integrates the Jacobian bound over tubes $T\mathbb{S}^2$3 around $T\mathbb{S}^2$4, splits the unit normal sphere into regions according to the sign of the smallest principal curvature $T\mathbb{S}^2$5, and observes that directions with $T\mathbb{S}^2$6 contribute only lower-order terms in $T\mathbb{S}^2$7 (using $T\mathbb{S}^2$8 there). Dividing by $T\mathbb{S}^2$9 and passing to the limit extracts the AVR. The necessity direction is a delicate bootstrap: equality forces $\mathbb{S}^2$0 on the positive-curvature cone $\mathbb{S}^2$1, forces the exact Jacobian identity pointwise, hence (via rigidity in the comparison theorem) constant ambient sectional curvature along radial geodesics, and ultimately that $\mathbb{S}^2$2 is an open hemisphere and the image of $\mathbb{S}^2$3 is a line — after which the pulled-back metric is computed explicitly. Notably, the equality case here does not assert that $\mathbb{S}^2$4 bounds a convex body or that $\mathbb{S}^2$5 is Euclidean; it characterizes equality through intrinsic conditions on $\mathbb{S}^2$6 and the normal exponential map.

Willmore–Chen inequality and absence of closed minimal submanifolds

Theorem (Willmore–Chen type). Let $\mathbb{S}^2$7, $\mathbb{S}^2$8, satisfy $\mathbb{S}^2$9 and Euclidean volume growth, and let $k$0 be an isometric immersion of a closed manifold. Then

$k$1

Equality holds if and only if $k$2 (where $k$3) is a diffeomorphism with pulled-back metric $k$4. In particular, equality implies $k$5 is an embedding, $k$6 is connected, totally umbilic with $k$7-parallel mean curvature vector, and $k$8.

The proof parallels the Chern–Lashof case but is simpler because only the mean curvature enters: splitting the normal sphere into $k$9 and its complement, the complement contributes lower-order terms since $$\mathrm{Ric}^{M}_{k}(v,P):=\sum_{i=1}^{k}\bar R(v,e_i,e_i,v),$$0 there, while on $$\mathrm{Ric}^{M}_{k}(v,P):=\sum_{i=1}^{k}\bar R(v,e_i,e_i,v),$$1 the leading term produces exactly $$\mathrm{Ric}^{M}_{k}(v,P):=\sum_{i=1}^{k}\bar R(v,e_i,e_i,v),$$2 per point after integration over the hemisphere. The rigidity analysis shows umbilicity and $$\mathrm{Ric}^{M}_{k}(v,P):=\sum_{i=1}^{k}\bar R(v,e_i,e_i,v),$$3 via a Codazzi computation combined with the symmetry and semidefiniteness of the tensor $$\mathrm{Ric}^{M}_{k}(v,P):=\sum_{i=1}^{k}\bar R(v,e_i,e_i,v),$$4 implied by $$\mathrm{Ric}^{M}_{k}(v,P):=\sum_{i=1}^{k}\bar R(v,e_i,e_i,v),$$5; the hypothesis $$\mathrm{Ric}^{M}_{k}(v,P):=\sum_{i=1}^{k}\bar R(v,e_i,e_i,v),$$6 is essential in forcing $$\mathrm{Ric}^{M}_{k}(v,P):=\sum_{i=1}^{k}\bar R(v,e_i,e_i,v),$$7.

An immediate corollary is a rigidity statement with no Euclidean analogue in content: if $$\mathrm{Ric}^{M}_{k}(v,P):=\sum_{i=1}^{k}\bar R(v,e_i,e_i,v),$$8 has nonnegative $$\mathrm{Ric}^{M}_{k}(v,P):=\sum_{i=1}^{k}\bar R(v,e_i,e_i,v),$$9-Ricci curvature and Euclidean volume growth ($\mathrm{AVR}(M,\bar g)>0$00), then $\mathrm{AVR}(M,\bar g)>0$01 admits no closed minimal submanifold of dimension $\mathrm{AVR}(M,\bar g)>0$02 for any $\mathrm{AVR}(M,\bar g)>0$03, since minimality would give $\mathrm{AVR}(M,\bar g)>0$04. This is a strong topological/geometric obstruction derived purely from the integral inequality.

Limitations and open questions

Several restrictions are acknowledged or evident. First, the Chern–Lashof inequality requires a sectional curvature lower bound ($\mathrm{AVR}(M,\bar g)>0$05), not merely $\mathrm{AVR}(M,\bar g)>0$06; whether the weaker hypothesis suffices there remains open. Second, the authors concede they could not weaken Heintze–Karcher's sectional condition using the original method, and the further comparison-geometric consequences of their theorem (obtainable by re-examining the Heintze–Karcher proof) are asserted but not proven. Third, in the hypersurface Willmore–Chen case of Agostiniani–Fogagnolo–Mazzieri, equality does not determine the interior of $\mathrm{AVR}(M,\bar g)>0$07, and it remains unknown whether equality forces $\mathrm{AVR}(M,\bar g)>0$08; the same question persists for the higher-codimension results here, whose equality cases are characterized intrinsically rather than by classification of the ambient space. Fourth, the sufficiency proof of the Willmore–Chen rigidity is omitted as "very similar" to the Chern–Lashof case, and several supporting lemmas are proven "by similar arguments," so the reader must reconstruct some details. Finally, the examples collected show that the relations among $\mathrm{AVR}(M,\bar g)>0$09, $\mathrm{AVR}(M,\bar g)>0$10, and $\mathrm{AVR}(M,\bar g)>0$11 are genuinely complicated, indicating that any refinement of the tube method must contend with cut-locus pathologies for non-embedded immersions.

Conclusion

The paper contributes three linked results: an explicit Jacobian decomposition for the normal exponential map valid in general ambient manifolds, a comparison theorem that replaces sectional curvature hypotheses by $\mathrm{AVR}(M,\bar g)>0$12-Ricci bounds in the higher-codimension Heintze–Karcher framework, and sharp Fenchel–Borsuk–Chern–Lashof and Willmore–Chen inequalities with full equality characterization on complete noncompact manifolds of nonnegative (respectively sectional / $\mathrm{AVR}(M,\bar g)>0$13-intermediate Ricci) curvature with Euclidean volume growth. The constants involve $\mathrm{AVR}(M,\bar g)>0$14, recovering the Euclidean inequalities when $\mathrm{AVR}(M,\bar g)>0$15, and the corollary excluding closed minimal submanifolds of intermediate dimension demonstrates that the inequalities carry genuine geometric force beyond their Euclidean prototypes.