# Quantitative stratification and the regularity of harmonic maps and minimal currents

Jeff Cheeger, Aaron Naber

Research output: Contribution to journalArticle

### Abstract

We provide techniques for turning estimates on the infinitesimal behavior of solutions to nonlinear equations (statements concerning tangent cones and their consequences) into more effective control. In the present paper, we focus on proving regularity theorems for minimizing harmonic maps and minimal currents. There are several aspects to our improvements of known estimates. First, we replace known estimates on the Hausdorff dimension of singular sets by estimates on the volumes of their r-tubular neighborhoods. Second, we give improved regularity control with respect to the number of derivatives bounded and/or on the norm in which the derivatives are bounded. As an example of the former, our results for minimizing harmonic maps $f: M^n \rightarrow N^m$ between Riemannian manifolds include a priori bounds in $W^{1,p} \cap W^{2,{p}/{2}}$ for all p < 3. These are the first such bounds involving second derivatives in general dimensions. Finally, the quantity we control actually provides much stronger information than that which follows from a bound on the Lp norm of derivatives. Namely, we obtain Lp bounds for the reciprocal of the regularity scale $r_f(x):= \max\{r: \sup_{B_r(x)}r|\nabla f|+r^2|\nabla^2 f|\leq 1\}$. Applications to minimal hypersufaces include a priori Lp bounds for the second fundamental form A for all p < 7. Previously known bounds were for $p < 4+ \sqrt{{8}/{n}}$ in the smooth immersed stable case. Again, the full theorem is much stronger and yields Lp bounds for the reciprocal of the corresponding regularity scale $r_{|A|}(x):= \max\{r: \sup_{B_r(x)}r|A|\leq 1\}$. In outline, our discussion follows that of an earlier paper in which we proved analogous estimates in the context of noncollapsed Riemannian manifolds with a lower bound on Ricci curvature. These were applied to Einstein manifolds. A key role in each of these arguments is played by the relevant quantitative differentiation theorem.

Original language English (US) 965-990 26 Communications on Pure and Applied Mathematics 66 6 https://doi.org/10.1002/cpa.21446 Published - Jun 2013

### Fingerprint

Harmonic Maps
Stratification
Regularity
Derivatives
Estimate
Theorem proving
Derivative
Riemannian Manifold
Theorem
Nonlinear equations
Cones
Tangent Cone
Einstein Manifold
Singular Set
A Priori Bounds
Second Fundamental Form
Lp-norm
Ricci Curvature
Behavior of Solutions
Second derivative

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### Cite this

In: Communications on Pure and Applied Mathematics, Vol. 66, No. 6, 06.2013, p. 965-990.

Research output: Contribution to journalArticle

@article{1bc88210fdd04c238a7c4d4d15ba9fec,
title = "Quantitative stratification and the regularity of harmonic maps and minimal currents",
abstract = "We provide techniques for turning estimates on the infinitesimal behavior of solutions to nonlinear equations (statements concerning tangent cones and their consequences) into more effective control. In the present paper, we focus on proving regularity theorems for minimizing harmonic maps and minimal currents. There are several aspects to our improvements of known estimates. First, we replace known estimates on the Hausdorff dimension of singular sets by estimates on the volumes of their r-tubular neighborhoods. Second, we give improved regularity control with respect to the number of derivatives bounded and/or on the norm in which the derivatives are bounded. As an example of the former, our results for minimizing harmonic maps $f: M^n \rightarrow N^m$ between Riemannian manifolds include a priori bounds in $W^{1,p} \cap W^{2,{p}/{2}}$ for all p < 3. These are the first such bounds involving second derivatives in general dimensions. Finally, the quantity we control actually provides much stronger information than that which follows from a bound on the Lp norm of derivatives. Namely, we obtain Lp bounds for the reciprocal of the regularity scale $r_f(x):= \max\{r: \sup_{B_r(x)}r|\nabla f|+r^2|\nabla^2 f|\leq 1\}$. Applications to minimal hypersufaces include a priori Lp bounds for the second fundamental form A for all p < 7. Previously known bounds were for $p < 4+ \sqrt{{8}/{n}}$ in the smooth immersed stable case. Again, the full theorem is much stronger and yields Lp bounds for the reciprocal of the corresponding regularity scale $r_{|A|}(x):= \max\{r: \sup_{B_r(x)}r|A|\leq 1\}$. In outline, our discussion follows that of an earlier paper in which we proved analogous estimates in the context of noncollapsed Riemannian manifolds with a lower bound on Ricci curvature. These were applied to Einstein manifolds. A key role in each of these arguments is played by the relevant quantitative differentiation theorem.",
author = "Jeff Cheeger and Aaron Naber",
year = "2013",
month = "6",
doi = "10.1002/cpa.21446",
language = "English (US)",
volume = "66",
pages = "965--990",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",
number = "6",

}

TY - JOUR

T1 - Quantitative stratification and the regularity of harmonic maps and minimal currents

AU - Cheeger, Jeff

AU - Naber, Aaron

PY - 2013/6

Y1 - 2013/6

N2 - We provide techniques for turning estimates on the infinitesimal behavior of solutions to nonlinear equations (statements concerning tangent cones and their consequences) into more effective control. In the present paper, we focus on proving regularity theorems for minimizing harmonic maps and minimal currents. There are several aspects to our improvements of known estimates. First, we replace known estimates on the Hausdorff dimension of singular sets by estimates on the volumes of their r-tubular neighborhoods. Second, we give improved regularity control with respect to the number of derivatives bounded and/or on the norm in which the derivatives are bounded. As an example of the former, our results for minimizing harmonic maps $f: M^n \rightarrow N^m$ between Riemannian manifolds include a priori bounds in $W^{1,p} \cap W^{2,{p}/{2}}$ for all p < 3. These are the first such bounds involving second derivatives in general dimensions. Finally, the quantity we control actually provides much stronger information than that which follows from a bound on the Lp norm of derivatives. Namely, we obtain Lp bounds for the reciprocal of the regularity scale $r_f(x):= \max\{r: \sup_{B_r(x)}r|\nabla f|+r^2|\nabla^2 f|\leq 1\}$. Applications to minimal hypersufaces include a priori Lp bounds for the second fundamental form A for all p < 7. Previously known bounds were for $p < 4+ \sqrt{{8}/{n}}$ in the smooth immersed stable case. Again, the full theorem is much stronger and yields Lp bounds for the reciprocal of the corresponding regularity scale $r_{|A|}(x):= \max\{r: \sup_{B_r(x)}r|A|\leq 1\}$. In outline, our discussion follows that of an earlier paper in which we proved analogous estimates in the context of noncollapsed Riemannian manifolds with a lower bound on Ricci curvature. These were applied to Einstein manifolds. A key role in each of these arguments is played by the relevant quantitative differentiation theorem.

AB - We provide techniques for turning estimates on the infinitesimal behavior of solutions to nonlinear equations (statements concerning tangent cones and their consequences) into more effective control. In the present paper, we focus on proving regularity theorems for minimizing harmonic maps and minimal currents. There are several aspects to our improvements of known estimates. First, we replace known estimates on the Hausdorff dimension of singular sets by estimates on the volumes of their r-tubular neighborhoods. Second, we give improved regularity control with respect to the number of derivatives bounded and/or on the norm in which the derivatives are bounded. As an example of the former, our results for minimizing harmonic maps $f: M^n \rightarrow N^m$ between Riemannian manifolds include a priori bounds in $W^{1,p} \cap W^{2,{p}/{2}}$ for all p < 3. These are the first such bounds involving second derivatives in general dimensions. Finally, the quantity we control actually provides much stronger information than that which follows from a bound on the Lp norm of derivatives. Namely, we obtain Lp bounds for the reciprocal of the regularity scale $r_f(x):= \max\{r: \sup_{B_r(x)}r|\nabla f|+r^2|\nabla^2 f|\leq 1\}$. Applications to minimal hypersufaces include a priori Lp bounds for the second fundamental form A for all p < 7. Previously known bounds were for $p < 4+ \sqrt{{8}/{n}}$ in the smooth immersed stable case. Again, the full theorem is much stronger and yields Lp bounds for the reciprocal of the corresponding regularity scale $r_{|A|}(x):= \max\{r: \sup_{B_r(x)}r|A|\leq 1\}$. In outline, our discussion follows that of an earlier paper in which we proved analogous estimates in the context of noncollapsed Riemannian manifolds with a lower bound on Ricci curvature. These were applied to Einstein manifolds. A key role in each of these arguments is played by the relevant quantitative differentiation theorem.

UR - http://www.scopus.com/inward/record.url?scp=84876040386&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84876040386&partnerID=8YFLogxK

U2 - 10.1002/cpa.21446

DO - 10.1002/cpa.21446

M3 - Article

AN - SCOPUS:84876040386

VL - 66

SP - 965

EP - 990

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 6

ER -