Minimal Surfaces | Gaoming Wang

The stable Bernstein theorem in $\mathbb{R}^{7}$

Mon, 14 Sep 2026 15:19:59 +0000

Main theorem

Stable Bernstein theorem

Flatness in $\mathbb R^7$

Every smooth, connected, complete, two-sided stable minimal immersion $F:M^6\to\mathbb R^7$ has $A\equiv0$, and its image is a hyperplane. No properness, volume-growth, or curvature bound is assumed.

Proof architecture

The proof first establishes all subcritical Green-function moments, then uses critical energy identities and the terminal Green flux to force flatness. Write $A$ for the second fundamental form and $G$ for the minimal positive Green function with a unit pole. For a fixed small $t_0>0$, set

$$ J(t)=\int_{{G=t}}|\nabla G|^3,dA_g, \qquad \mathcal M_\sigma=\int_0^{t_0}J(t)t^{-\sigma-1},dt. $$

Part I · All subcritical moments

Step 1 · Preparation

Geometric reduction and Green functions

A blow-up argument reduces the problem to bounded curvature. Unit Green flux and the estimate $|\nabla G|\leq CG$ give the initial integrability range $\mathcal M_\sigma<\infty$ for $\sigma<2$. The target is every $\sigma<5/2$, the critical exponent in dimension six.

Step 2 · Initial improvement

From $2$ to $2.46$

Bochner's formula, Simons' identity, stability, and the divergence-free tensor $\tfrac12|A|^2g-A^2$ supply the integral identities. The full Codazzi constraint sharpens the directional estimates, while a Schoen–Simon–Yau inequality retaining its Laplacian term controls cutoff errors. Moment continuation yields $\mathcal M_{2.46}<\infty$.

Step 3 · Nonlinear stability

From $2.46$ to $2.4962$

Apply stability to a nonlinear norm combining a curvature test function and a Green-gradient test function. Differentiating this norm retains an extra nonnegative gradient term. Together with the Green and curvature identities, it gives coercive estimates on fifteen rational parameter intervals and proves $\mathcal M_{2.4962}<\infty$.

Step 4 · Spectral coercivity

Every exponent below $5/2$

Additional divergence identities for homogeneous curvature tensors involving $A^4$, $\operatorname{tr}(A^3)A$, and $\operatorname{tr}(A^4)g$ strengthen the coercive estimate near the critical exponent. They continue the moment range from $2.4962$ to every $\sigma<5/2$.

Part II · Critical energies and rigidity

Step 5 · Critical energies

Remove the exterior cutoffs

All subcritical moments are now finite. An elementary spectral inequality, stability, and Simons' identity give finite critical curvature and derivative energies. The argument uses each fixed subcritical moment, with no uniform bound required as $\sigma\uparrow5/2$.

Step 6 · Rigidity

Terminal flux forces flatness

The energy bounds give finite limits of the normalized Green flux $t^{-5/2}J(t)$ at both ends. Brendle's sharp isoperimetric inequality determines the sign of the terminal Bochner boundary term. Combining this sign with the critical spectral inequality forces the critical curvature energy to vanish, so $A\equiv0$.

Mixed Radial Volume Comparison under Spectral Ricci Bounds

Wed, 02 Sep 2026 00:00:00 +0000

Main results

This paper develops radial volume comparison under a spectral lower bound for Ricci curvature. The mixed radial construction separates the conformal geometry used to choose rays from the weights used to truncate and measure them.

Sharp comparison

A space-form model for mixed radial balls

Under the spectral inequality $\operatorname{Ric}_g-\alpha(\Delta_g u/u)g\geq(n-1)\kappa g$, the weighted volume of the $g$-arclength mixed radial domain satisfies a scale-invariant comparison with the corresponding space-form volume.

Polynomial growth

No pointwise bounds on the spectral weight

For an explicit range of radial and measure exponents, mixed radial balls have polynomial weighted-volume growth. The optimal exponent can attain the Euclidean value $n$.

Applications

Diameter, sharp volume, and stable Bernstein

The comparison gives radial proofs of the spectral Bonnet--Myers and sharp volume theorem of Antonelli--Xu, together with a short volume-growth proof that every complete two-sided stable minimal immersion $M^3\to\mathbb R^4$ is flat.

Proof architecture

The argument begins with an exact Riccati identity for the weighted radial Jacobian along geodesics of the conformal metric $h_\alpha=u^{2\alpha}g$. Taking the $(n-1)$-st root of this Jacobian converts the identity into a one-dimensional Sturm comparison and yields the space-form estimate. For general mixed radial balls, the logarithmic derivatives of the Jacobian and the spectral weight are instead treated as coupled variables; a quadratic estimate and a Bellman-type correction produce the polynomial-growth monotonicity formula. The positive-curvature case also gives an optimal diameter bound among smooth closed examples.

Stable Minimal Hypersurfaces in Positively Curved $4$-Manifolds

Sun, 23 Aug 2026 07:45:35 +0000

Main results

This paper studies complete two-sided stable minimal hypersurfaces of dimension three in positively curved four-manifolds. It identifies the curvature gap that forces rigidity and shows, by an explicit construction, why that gap is essential.

Rigidity

Positive scalar curvature forces total geodesicity

If the ambient sectional curvature is nonnegative and the ambient scalar curvature satisfies $\overline R\geq\kappa>0$, then every complete, connected, two-sided stable minimal immersion $M^3\to X^4$ has $A\equiv0$ and $\overline{\operatorname{Ric}}(\nu,\nu)\equiv0$.

Sharpness

The uniform scalar curvature gap is necessary

There is a complete metric with strictly positive sectional curvature on $\mathbb R^4$ containing a complete, embedded, one-ended, nonparabolic, two-sided stable minimal hypersurface $\Sigma^3\cong\mathbb R^3$ that is not totally geodesic.

Improvement

No bounded-geometry or upper-curvature hypothesis

The rigidity theorem removes the weakly bounded geometry assumption from the earlier result of Chodosh--Li--Stryker. It also requires no uniform upper bound for the ambient sectional curvature.

Proof architecture

The argument replaces extrinsic volume comparison by an intrinsic spectral and potential-theoretic route.

Step 1

Spectral and topological reduction

A positive Jacobi function turns stability into spectral Ricci and scalar curvature inequalities. Sharp splitting and topology theorems reduce the only nontrivial case to a one-ended manifold diffeomorphic to $\mathbb R^3$.

Step 2

Separating warped $\mu$-bubbles

Escaping warped $\mu$-bubble separators are constructed with a uniform integral estimate for their mean curvature, the prescribed curvature term, and the normal derivative of the Jacobi weight.

Step 3

Harmonic functions and flux

A level-set identity gives a uniform $L^1$ bound for capacitor gradients. Nonparabolicity would contradict conservation of flux, so the hypersurface is parabolic; stability cutoffs then force the Jacobi potential to vanish identically.

The counterexample mechanism

Starting from the Chodosh–Li–Stryker positively curved model, the ambient metric is changed only near a compact subset of a totally geodesic stable hypersurface. The perturbation introduces a nonzero trace-free second fundamental form while preserving both the induced metric and the Jacobi potential

$$ |A|^2+\overline{\operatorname{Ric}}(\nu,\nu). $$

Thus completeness, minimality, and stability remain unchanged, while the hypersurface ceases to be totally geodesic. Because the deformation is compactly supported and small in $C^2$, strict positivity of sectional curvature is preserved.

Regularity of branched stable minimal immersed hypersurfaces

Fri, 31 Jul 2026 17:09:37 +0000

Main results

This work develops a regularity theory for branched stable minimal immersed hypersurfaces. The conclusions are sharp, and the examples show that the remaining singularities are a genuine feature of the immersed setting rather than an artifact of the proof.

Sharp regularity

An optimal bound for non-branch singularities

The non-branch singular set is empty in dimension $n=2$, discrete in dimension $n=3$, and has Hausdorff dimension at most $n-3$ for every $n\geq4$.

Sharpness examples

A stable cone with a genuine non-branch singularity

We construct a non-flat stable minimal cone in $\mathbb R^4$ arising from a branched minimal immersion. Its products with Euclidean factors attain the $n-3$ bound in every dimension $n\geq3$.

Analytic mechanism

A branched sheeting theory

The proof combines a generalized Schoen inequality with a sheeting theorem near stationary classical cones and unions of hyperplanes.

Starting model for the construction

The construction of our sharpness examples starts from a minimal surface in $\mathbb S^3$ obtained by desingularizing two orthogonal Clifford tori. The image below visualizes this starting model, which provides the geometric input for the subsequent construction of a branched stable minimal cone in $\mathbb R^4$.

Starting model for the sharpness examples: a numerical visualization of the $m=30$ Choe–Soret surface in $\mathbb S^3$, shown in stereographic projection. This depicts the starting surface, not the final branched stable cone. The bronze finish and pedestal are presentation elements.

Companion materials

The two resources below give a closer view of how the project developed and of the geometry behind one of its principal examples.

Research record

How we found the test function

This note follows the route from the capillary prototype through numerical screening, the small-$k$ expansion, the local linear repair, and the cutoff ODE. GitHub Copilot and Cursor Composer 2 were used chiefly to turn hand calculations into rapidly testable code; the numerical evidence guided the choice of formula, while the final verification remained analytic.

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Interactive geometry

Desingularization of two orthogonal Clifford tori

An interactive 3D viewer provides a geometric preview of the desingularization: rotate and zoom the model, or isolate individual regions to inspect how the two tori are joined.

Open the 3D viewer

Spectral splitting theorem and ends of minimal hypersurfaces

Thu, 14 May 2026 15:06:46 +0000

Main results

The paper develops a geodesic-line approach to spectral Ricci curvature and uses it to control the topology at infinity of minimal hypersurfaces.

Spectral splitting

A weighted minimizing line recovers classical rigidity

If $\lambda_1(-\alpha\Delta+\operatorname{Ric})\geq0$ with $\alpha<4/(n-1)$ and the associated weighted metric admits a minimizing line, then $\operatorname{Ric}\geq0$ pointwise and the manifold splits off an $\mathbb R$ factor.

Ends at infinity

Infinitely many ends create a weighted line

Nonnegative spectral Ricci curvature outside a compact set, together with infinitely many ends, forces the existence of a weighted minimizing geodesic line outside a larger compact set.

Minimal hypersurfaces

Finite index implies finitely many ends

Every complete two-sided finite-index minimal hypersurface in an ambient manifold of dimension at most five with nonnegative biRic curvature has finitely many ends, without a properness assumption.

Proof strategy

The splitting proof adapts the Cheeger–Gromoll Busemann-function argument to the weighted length functional determined by a positive solution of the spectral equation. For the application, infinitely many ends are used to construct a weighted line at infinity; the positive solution is then perturbed to make the spectral inequality strict outside a compact set. The second variation of weighted length produces the contradiction and hence finiteness of ends.

Scalar curvature rigidity of domains in a 3-dimensional warped product

Thu, 06 Mar 2025 11:51:14 +0800

Main results

The paper establishes Llarull-type scalar-curvature rigidity for domains in three-dimensional spherical warped products, including models with boundary and singular endpoint geometry.

Domain rigidity

Metric, scalar, and boundary comparison determine the model

For a domain whose side boundary is convex in the conformally related product metric, the inequalities $g\geq\bar g$, $R_g\geq R_{\bar g}$, and $H_g\geq H_{\bar g}$ force $g=\bar g$ under the stated endpoint hypotheses.

Endpoint geometry

Smooth, conical, and disk-type ends are treated uniformly

The rigidity theorem covers smooth poles, Euclidean or metric tangent cones, and truncations by end disks with matching mean-curvature and angle inequalities.

Conical Llarull theorem

Antipodal cone points are allowed

For warped three-spheres with $\psi(t_\pm)=a_\pm|t-t_\pm|+o(|t-t_\pm|)$ and $0

Proof strategy

Prescribed-mean-curvature capillary $\mu$-bubbles play the role of the minimizing hypersurfaces in Llarull’s theorem. The stability inequality and a Gauss–Bonnet comparison make a minimizing bubble infinitesimally rigid, and a local foliation propagates equality through the domain. Near a vanishing warping factor or a conical endpoint, local barriers and tangent-cone analysis replace a formal singular perturbation and ensure that the minimizing problem cannot escape through the pole.

A splitting theorem for 3-manifold with nonnegative scalar curvature and mean-convex boundary

Thu, 16 Jan 2025 04:51:14 +0800

Main results

This work proves scalar-curvature splitting theorems for noncompact three-manifolds with mean-convex boundary, using noncompact minimizing free-boundary and capillary surfaces as the rigidifying objects.

Free-boundary splitting

A minimizing half-cylinder or strip forces flatness

If the manifold contains an absolutely area-minimizing free-boundary half-cylinder or strip, then it is flat. Up to scaling, the resulting models are covered by $\mathbb S^1\times\mathbb R^2_+$ or are isometric to $[0,1]\times\mathbb R^2$.

Capillary extension

The theorem holds for every contact angle

For $\theta\in(0,\pi)$, an absolutely $\theta$-energy-minimizing half-cylinder gives the corresponding tilted flat model; the strip statement also holds when its two boundary curves lie on different boundary components.

No compact-boundary hypothesis

The ambient boundary may be noncompact or disconnected

The argument requires neither compactness nor a prescribed number of components of $\partial M$, extending earlier boundary splitting theorems beyond their usual global assumptions.

Proof strategy

Plateau-type problems in carefully chosen prism-shaped domains produce a sequence of minimizing capillary surfaces near the given noncompact surface. Stability classifies the possible limits, while energy comparisons rule out unwanted disk, half-plane, plane, and cylinder degenerations. The surviving half-cylinders or strips form a rigid family; equality in the scalar-curvature and boundary inequalities then propagates to an ambient flat product.

On $\delta$-Stable Minimal Hypersurfaces in $\mathbb{R}^{n+1}$

Thu, 04 Jul 2024 11:51:14 +0800

Main results

The paper extends regularity, compactness, and Bernstein theory from stable minimal hypersurfaces to the weaker notion of $\delta$-stability.

Regularity and compactness

A quantitative theory above the catenoid threshold

For $n\geq3$ and $\delta>(n-2)/n$, sequences with uniform area control admit varifold subsequences whose singular sets satisfy an explicit dimension bound determined by $\delta$.

Bernstein theorem

Flatness in dimensions three and four

For $n=3,4$, complete two-sided $\delta$-stable immersed hypersurfaces are flat under either a stronger numerical stability range or the combination of simple connectivity, properness, and finitely many ends.

Sharp threshold

The lower bound on $\delta$ cannot be improved

The $n$-dimensional catenoid is exactly $(n-2)/n$-stable. Simons-type cones also identify the sharp threshold for the cone-classification component of the theory.

Proof strategy

The regularity argument combines a Caccioppoli inequality, De Giorgi iteration, an $\varepsilon$-regularity theorem for the second fundamental form, and tangent-cone analysis. A sharp classification of $\delta$-stable minimal cones supplies the dimension reduction. In low dimensions, point-picking and curvature estimates reduce the Bernstein statements to the global flatness results.

Allard-Type Regularity for Varifolds with Prescribed Contact Angle

Tue, 26 Mar 2024 11:51:14 +0800

Main results

This paper establishes a boundary Allard theorem for varifolds with prescribed contact angle, including variable angles and ambient metrics of low regularity.

Boundary regularity

A multiplicity-one tangent half-plane gives smooth structure

If a boundary point admits a multiplicity-one tangent half-hyperplane, then the support is locally a $C^{1,\gamma}$ hypersurface with boundary for some $\gamma\in(0,1)$.

Contact angle

The variational angle is recovered geometrically

On the regular support, the geometric contact angle is the prescribed angle (apart from the explicitly identified orthogonal alternative); the ambiguity disappears under a natural nonorthogonality condition.

Density criterion

Mass control alone suffices in favorable regimes

For two-dimensional varifolds, or in every dimension when the angle is sufficiently close to $90^\circ$, a quantitative boundary density bound implies the required regularity.

Proof strategy

The proof is carried out in a flattened half-space equipped with a $C^1$ metric, so it remains intrinsic to the codimension-one contact-angle problem. It combines a boundary monotonicity formula with an $L^2$ height and tilt estimate, a blow-up analysis, and geometric excess decay. A reflection idea inspired by Leon Simon explains how the boundary term and the capillary angle are incorporated into a stationary comparison object.

Dihedral rigidity in hyperbolic 3-space

Wed, 08 Nov 2023 11:51:14 +0800

Main results

The paper proves hyperbolic analogues of Gromov’s dihedral comparison and rigidity conjectures for tetrahedra and a broad class of prisms.

Comparison

All inequalities cannot be simultaneously strict

For the prescribed tetrahedra and prisms, scalar curvature greater than $-6$ and face mean curvature above the hyperbolic reference prevent every dihedral angle from being strictly smaller than its model value.

Rigidity

Weak comparison forces the hyperbolic model

Under the cylinder-trapping condition, or the stated side-angle alternative, the inequalities $R_g\geq-6$, $H_g\geq H_{\mathrm{hyp}}$, and $\gamma_g\leq\gamma_{\mathrm{hyp}}$ imply that the polyhedron is isometric to one in hyperbolic space.

Geometric scope

Every hyperbolic tetrahedron in the class is covered

All cone-type references satisfy cylinder trapping; for prisms it is equivalent to a concrete nesting condition between the two Euclidean base polygons.

Proof strategy

A variational functional produces a stable capillary surface of constant mean curvature $\pm2$ with the contact angles of the reference foliation. Its second variation, together with Gauss–Bonnet, converts scalar-, mean-, and dihedral-curvature comparisons into a contradiction unless equality holds. In the equality case, an infinitesimally rigid capillary surface is propagated by a constant-mean-curvature foliation; a separate foliation construction at a vertex completes the global rigidity argument.

Scalar curvature comparison of rotationally symmetric sets

Tue, 25 Apr 2023 11:51:14 +0800

Main results

This paper treats scalar-curvature rigidity for compact three-manifolds whose reference boundary is rotationally symmetric and weakly convex, including smooth, conical, and truncated models.

Euclidean rigidity

Boundary comparison determines the interior

If $R_g\geq0$, the induced boundary metric and mean curvature dominate those of the Euclidean reference, and the nonsmooth dihedral angles are no larger, then the manifold is flat under the stated pole hypotheses.

Endpoint geometries

Disks, cones, and spherical poles are all included

The theorem covers boundaries cut off by planar disks, boundaries with conical tips, and boundaries with smooth spherical poles. Standard balls and right circular cones are important special cases.

Extensions

Hyperbolic and circle-symmetric analogues

The same capillary framework yields corresponding rigidity statements in hyperbolic space and in nonpositively curved spaces with an $\mathbb S^1$ symmetry.

Proof strategy

The argument minimizes a capillary functional whose prescribed contact angle varies along the rotational boundary. Stability and Gauss–Bonnet force the minimizing disk to be infinitesimally rigid, after which a foliation propagates the equality. The main difficulty is ensuring that a nontrivial minimizer exists near a conical or spherical pole; local constant-mean-curvature barrier foliations reduce those cases to the planar-end problem.

Generalized Bernstein Theorem for Stable Minimal Plateau Surfaces

Thu, 20 Oct 2022 11:51:14 +0800

Main results

The classical Bernstein question is extended from smooth minimal surfaces to Plateau surfaces, whose local models may also contain $Y$- and $T$-type soap-film singularities.

Generalized Bernstein theorem

Stability and quadratic area growth force flatness

If a complete orientable minimal Plateau surface in $\mathbb R^3$ is stable and has at most quadratic area growth, then every component of its regular part lies in a plane.

No $T$-points

The surface is cylindrical over a planar network

When no $T$-type singularity occurs, the surface is, after a rigid motion, of the form $N\times\mathbb R$ for an embedded stationary network $N\subset\mathbb R^2$.

With $T$-points

Only two global configurations remain

If $T$-type singularities occur, there is either one such point and the surface is the standard $T$ cone, or there are exactly two points joined in the explicitly classified flat configuration.

Proof strategy

The proof transfers the stability inequality to the regular sheets while retaining the balancing information along the singular curves. Curvature estimates and logarithmic cutoffs, made possible by quadratic area growth, force the second fundamental form to vanish. Once each sheet is planar, the topology and Plateau angle conditions reduce the remaining work to a global classification of the straight $Y$-curves and isolated $T$-points.

General Theory of Partial Differential Equations on Triple Junction Surfaces

Sat, 02 Jul 2022 11:51:14 +0800

Main contributions

The thesis builds a general analytic framework for triple-junction hypersurfaces: singular spaces obtained by identifying the boundaries of several smooth sheets.

Function spaces

Analysis that remembers the junction geometry

Sobolev- and Hölder-type spaces are defined sheetwise with compatibility and balancing conditions along the common boundary, providing the correct domains for geometric variation problems.

Elliptic theory

Classical PDE tools extend to the singular setting

For second-order elliptic operators on triple-junction hypersurfaces, the thesis develops existence, regularity, Fredholm, and spectral results analogous to those on smooth Riemannian manifolds.

Applications

Geometry becomes accessible through the PDE framework

The theory is applied to Jacobi operators, Morse index questions for minimal triple junctions, and conformal and uniformization problems for singular surfaces.

Analytic viewpoint

The central principle is to separate local elliptic analysis on each smooth sheet from the finite-dimensional matching data along the junction. Trace conditions encode which functions arise from ambient variations, while the conormal balance supplies the natural transmission condition for second-order operators. Once regularity across that interface is established, variational, spectral, and geometric tools can be used much as in the smooth theory.

Curvature Estimates for Stable Minimal Surfaces with a Common Free Boundary

Thu, 19 May 2022 11:41:35 +0800

Main results

The paper extends curvature estimates and Bernstein-type rigidity from smooth minimal surfaces to weighted multiple-junction surfaces sharing a common free boundary.

Curvature estimate

An $L^p$ theory compatible with the junction

For stable minimal multiple-junction surfaces, the second fundamental forms satisfy an $L^{2p}$ estimate for $1

Compact junction

Quadratic growth forces every sheet to be flat

A complete, orientable, stable minimal multiple-junction surface in $\mathbb R^3$ with quadratic area growth, compact common boundary, and constant pairwise angles has flat sheets.

Straight junction

The same Bernstein conclusion holds along a line

For a triple junction meeting at $120^\circ$, the flatness conclusion also holds when the common boundary is a straight line, under the corresponding completeness, stability, and area-growth assumptions.

Proof strategy

Simons’ inequality is applied sheet by sheet and combined with the global stability inequality using test functions that satisfy the junction compatibility condition. The conormal boundary terms do not disappear automatically; the weighted balance law reorganizes them into a controllable expression. Iteration of the resulting $L^p$ estimate with large cutoffs and quadratic area growth forces the curvature to vanish.

Uniformization of surfaces with boundary and the application to the triple junction surfaces with negative Euler characteristic

Sat, 09 Oct 2021 11:51:14 +0800

Main results

The paper first develops a quantitative uniformization theorem for surfaces with boundary and then uses it to match the three sheets of a triple-junction surface.

Surfaces with boundary

Prescribed constant interior and boundary curvature

For a compact oriented surface with negative Euler characteristic and $(k,c)$ satisfying $k\leq0$ and $c<\sqrt{-k}$, there is a unique conformal metric with Gaussian curvature $k$ and constant boundary geodesic curvature $c$.

Quantitative control

Area and boundary length vary monotonically

The area $A(k,c)$ and boundary length $L(k,c)$ are continuous and satisfy sharp monotonicity and limiting laws, including the behavior of $cL(k,c)$ needed to balance junction curvature.

Triple junctions

A weak hyperbolic uniformization

If a triple-junction surface has connected junction and $\chi(M)\leq0$, its three sheets admit conformal hyperbolic metrics with a common boundary length whose geodesic curvatures sum to zero; the sheet metric is unique when its Euler characteristic is nonpositive.

Proof strategy

The prescribed-curvature problem is written as a semilinear elliptic boundary value problem. Maximum-principle estimates give existence, uniqueness, continuity, and monotonicity as $k$ and $c$ vary. For the junction problem, the boundary length is used as a common parameter: the three curvature-length functions are added, and their monotonicity and endpoint limits produce the unique balancing length.

Positive-curvature phenomenon

The following is a short video shows that why the uniqueness might fail when $K>0$.

In this video, we suppose the surface $\Sigma$ is the two copies of the domain enclosed by six arcs by identifying their corresponding edges marked with green color. So here $\Sigma$ is just a genus zero surface with three boundary components. We fix one boundary of $\Sigma$ having geodesic curvature $-1$ (marked as red) and the remaining two boundaries being geodesic (marked as black). Then we can let the area grow to see how the Gauss curvature changes after uniformization. The blue circle in this video represents the infinity circle if $K<0$ or the geodesic circle if $K>0$. From the video, we can see that $K$ will increase as area $A$ increases, then $K$ will reach its maximal point and after that $K$ will decrease to 0. So we will see that for some $K>0$, there are indeed two metrics with prescribed Gauss curvature and geodesic curvatures of boundaries.