L coefficient operators and non-smooth Riemannian metrics
|
|
- Clara Peters
- 5 years ago
- Views:
Transcription
1 L coefficient operators and non-smooth Riemannian metrics Lashi Bandara Centre for Mathematics and its Applications Australian National University 17 October 2012 Geometric Analysis Seminar Stanford University Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
2 Setup Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure dµ. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
3 Setup Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure dµ. Write div = in L 2 and let S = (I, ). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
4 Setup Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure dµ. Write div = in L 2 and let S = (I, ). Consider the following uniformly elliptic second order differential operator L A : D(L A ) L 2 (M) L 2 (M) defined by L A u = as ASu = a div(a 11 u) a div(a 10 u) + aa 01 u + aa 00 u. where a and A = (A ij ) are L multiplication operators. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
5 Setup Let M be a smooth, complete Riemannian manifold with metric g, Levi-Civita connection, and volume measure dµ. Write div = in L 2 and let S = (I, ). Consider the following uniformly elliptic second order differential operator L A : D(L A ) L 2 (M) L 2 (M) defined by L A u = as ASu = a div(a 11 u) a div(a 10 u) + aa 01 u + aa 00 u. where a and A = (A ij ) are L multiplication operators. That is, that there exist κ 1, κ 2 > 0 such that Re av, v κ 1 v 2, v L 2 Re ASu, Su κ 2 ( u 2 + u 2 ), u H 1 Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
6 The problem The Kato square root problem on manifolds is to determine when the following holds: Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
7 The problem The Kato square root problem on manifolds is to determine when the following holds: { D( LA ) = H 1 (M) L A u u + u = u H 1, u H 1 (M) Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
8 Theorem (B.-Mc [BMc]) Let (M, g) be a smooth, complete Riemannian manifold Ric C and inj(m) κ > 0. Suppose there exist κ 1, κ 2 > 0 such that Re av, v κ 1 v 2 Re ASu, Su κ 2 u 2 H 1 for v L 2 (M) and u H 1 (M). Then, D( L A ) = D( ) = H 1 (M) and L A u u + u = u H 1 for all u H 1 (M). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
9 Stability Theorem (B.-Mc [BMc]) Let (M, g) be a smooth, complete Riemannian manifold with Ric C and inj(m) κ > 0. Suppose that there exist κ 1, κ 2 > 0 such that Re av, v κ 1 v 2 Re ASu, Su κ 2 u 2 H 1 for v L 2 (M) and u H 1 (M). Then for every η i < κ i, whenever ã η 1, Ã η 2, the estimate L A u L A+ Ã u ( ã + Ã ) u H 1 holds for all u H 1 (M). The implicit constant depends in particular on A, a and η i. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
10 History of the problem In the 1960 s, Kato considered the following abstract evolution equation du dt on a Hilbert space H. + A(t)u = f(t), t [0, T ]. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
11 History of the problem In the 1960 s, Kato considered the following abstract evolution equation du dt on a Hilbert space H. + A(t)u = f(t), t [0, T ]. In 1962, Kato showed in [Kato61] that for 0 α < 1/2 and A(t) maximal accretive that D(A(t) α ) = D(A(t) α ) = D α = const, and A(t) α u A(t) α u, u D α. (K α ) Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
12 History of the problem In the 1960 s, Kato considered the following abstract evolution equation du dt on a Hilbert space H. + A(t)u = f(t), t [0, T ]. In 1962, Kato showed in [Kato61] that for 0 α < 1/2 and A(t) maximal accretive that D(A(t) α ) = D(A(t) α ) = D α = const, and A(t) α u A(t) α u, u D α. (K α ) Counter examples were known for α > 1/2. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
13 Kato asked two questions: (K1) Does (K α ) hold for α = 1/2? Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
14 Kato asked two questions: (K1) Does (K α ) hold for α = 1/2? (K2) For the case ω = 0, we know (K1) is automatically true, but is d A(t)u u dt for u W? Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
15 Kato asked two questions: (K1) Does (K α ) hold for α = 1/2? (K2) For the case ω = 0, we know (K1) is automatically true, but is d A(t)u u dt for u W? In 1972, McIntosh provided a counter example in [Mc72] demonstrating that (K1) is false in such generality. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
16 Kato asked two questions: (K1) Does (K α ) hold for α = 1/2? (K2) For the case ω = 0, we know (K1) is automatically true, but is d A(t)u u dt for u W? In 1972, McIntosh provided a counter example in [Mc72] demonstrating that (K1) is false in such generality. In 1982, McIntosh showed that (K2) also did not hold in general in [Mc82]. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
17 The Kato square root problem then became the following. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
18 The Kato square root problem then became the following. Suppose A L is a pointwise matrix multiplication operator satisfying the following ellipticity condition: Re A u, u κ u 2, for some κ > 0. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
19 The Kato square root problem then became the following. Suppose A L is a pointwise matrix multiplication operator satisfying the following ellipticity condition: Re A u, u κ u 2, for some κ > 0. Is it then true that D( div A ) = H 1 (R n ) div A u u (K1) This was answered in the positive in 2002 by Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh and Phillipe Tchamitchian in [AHLMcT]. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
20 Axelsson (Rosén)-Keith-McIntosh framework In [AKMc], the authors created a first-order framework using the language of Dirac type operators to study such problems. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
21 Axelsson (Rosén)-Keith-McIntosh framework In [AKMc], the authors created a first-order framework using the language of Dirac type operators to study such problems. (H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert space H, Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
22 Axelsson (Rosén)-Keith-McIntosh framework In [AKMc], the authors created a first-order framework using the language of Dirac type operators to study such problems. (H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert space H, (H2) Suppose that B 1, B 2 L(H ) such that here exist κ 1, κ 2 > 0 satisfying Re B 1 u, u κ 1 u 2 and Re B 2 v, v κ 2 v 2 for u R(Γ ) and v R(Γ), Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
23 Axelsson (Rosén)-Keith-McIntosh framework In [AKMc], the authors created a first-order framework using the language of Dirac type operators to study such problems. (H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert space H, (H2) Suppose that B 1, B 2 L(H ) such that here exist κ 1, κ 2 > 0 satisfying Re B 1 u, u κ 1 u 2 and Re B 2 v, v κ 2 v 2 for u R(Γ ) and v R(Γ), (H3) The operators B 1, B 2 satisfy B 1 B 2 R(Γ) N (Γ) and B 2 B 1 R(Γ ) N (Γ ). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
24 Axelsson (Rosén)-Keith-McIntosh framework In [AKMc], the authors created a first-order framework using the language of Dirac type operators to study such problems. (H1) Let Γ be a densely-defined, closed, nilpotent operator on a Hilbert space H, (H2) Suppose that B 1, B 2 L(H ) such that here exist κ 1, κ 2 > 0 satisfying Re B 1 u, u κ 1 u 2 and Re B 2 v, v κ 2 v 2 for u R(Γ ) and v R(Γ), (H3) The operators B 1, B 2 satisfy B 1 B 2 R(Γ) N (Γ) and B 2 B 1 R(Γ ) N (Γ ). Let Γ B = B 1Γ B 2, Π B = Γ + Γ B and Π = Γ + Γ. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
25 Quadratic estimates and Kato type problems Proposition If (H1)-(H3) are satisfied and ˆ 0 tπ B (I + t 2 Π 2 B) 1 u 2 dt t u for u R(Π B ), then (i) D(Γ) D(Γ B ) = D(Π B) = D( Π 2 B ), and (ii) Γu + Γ B u Π B u Π 2 B u, for all u D(Π B). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
26 Set H = L 2 (M) (L 2 (M) L 2 (T M)). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
27 Set H = L 2 (M) (L 2 (M) L 2 (T M)). Define Γ g = [ ] 0 0, B S 0 1 = [ ] a 0, and B = [ ] A Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
28 Set H = L 2 (M) (L 2 (M) L 2 (T M)). Define Γ g = [ ] 0 0, B S 0 1 = [ ] a 0, and B = [ ] A Then, [ ] Γ 0 S g = 0 0 and Π 2 B = [ ] LA 0 0 Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
29 Set H = L 2 (M) (L 2 (M) L 2 (T M)). Define Γ g = [ ] 0 0, B S 0 1 = [ ] a 0, and B = [ ] A Then, [ ] Γ 0 S g = 0 0 and Π 2 B = [ ] LA 0 0 and Π B,g (u, 0) = (0, u, u) and Π 2 B,g (u, 0) = ( L A u, 0). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
30 Theorem 1 and 2 are a direct consequence of the following. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
31 Theorem 1 and 2 are a direct consequence of the following. Proposition Let M be a smooth, complete manifold with smooth metric g. Suppose there exist η, κ > 0 such that Ric g η and inj(m, g) κ. Furthermore, suppose that there exist κ 1, κ 2 such that Re av, v κ 1 v 2, v L 2 (M, g) Re ASu, Su κ 2 u 2 H 1, u H1 (M, g). Then, for u R(Π B ). ˆ 0 tπ B (I + t 2 Π 2 B) 1 u 2 dt t u 2 Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
32 Rough metrics We let M be a smooth, complete manifold as before but now let g be a C 0 metric. Let µ g denote the volume measure with respect to g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
33 Rough metrics We let M be a smooth, complete manifold as before but now let g be a C 0 metric. Let µ g denote the volume measure with respect to g. Definition Let h C 0 (T (2,0) M). Then, define h op,g = esssup µg(x) a.e. sup h x (u, v), u g = v g =1 and h op,s,g = sup x M sup h x (u, v). u g = v g =1 Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
34 Rough metrics We let M be a smooth, complete manifold as before but now let g be a C 0 metric. Let µ g denote the volume measure with respect to g. Definition Let h C 0 (T (2,0) M). Then, define h op,g = esssup µg(x) a.e. sup h x (u, v), u g = v g =1 and h op,s,g = sup x M sup h x (u, v). u g = v g =1 We say that two C 0 metrics g and g are δ-close if g g op,g < δ for δ > 0. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
35 Translating from one metric to another Suppose that we now have a smooth metric g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
36 Translating from one metric to another Suppose that we now have a smooth metric g. Write f u (v) = g(u, v) g(u, v). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
37 Translating from one metric to another Suppose that we now have a smooth metric g. Write f u (v) = g(u, v) g(u, v). By Riesz-Representation theorem, we find that there exists B such that g((i + B)u, v) = g(u, v). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
38 Translating from one metric to another Suppose that we now have a smooth metric g. Write f u (v) = g(u, v) g(u, v). By Riesz-Representation theorem, we find that there exists B such that g((i + B)u, v) = g(u, v). That is, exactly, we can absorb the lack of regularity of g in terms of a symmetric, bounded operator in g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
39 Translating from one metric to another Suppose that we now have a smooth metric g. Write f u (v) = g(u, v) g(u, v). By Riesz-Representation theorem, we find that there exists B such that g((i + B)u, v) = g(u, v). That is, exactly, we can absorb the lack of regularity of g in terms of a symmetric, bounded operator in g. The operator B = g g op,g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
40 Translating from one metric to another Suppose that we now have a smooth metric g. Write f u (v) = g(u, v) g(u, v). By Riesz-Representation theorem, we find that there exists B such that g((i + B)u, v) = g(u, v). That is, exactly, we can absorb the lack of regularity of g in terms of a symmetric, bounded operator in g. The operator B = g g op,g. The change of measures is given in terms of I + B. θ = dµ g dµ g Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
41 Stability of function spaces Proposition Let g and g be two C 0 metrics on M. If there exists δ [0, 1) such that g g op,g < δ, then (i) the spaces L 2 (M, g) = L 2 (M, g) with (1 δ) n 4 2,g 2, g (1 + δ) n 4 2,g, (ii) the Sobolev spaces H 1 (M, g) = H 1 (M, g) with (1 δ) n δ H 1,g H 1, g (1 + δ) n 4 1 δ H 1,g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
42 Π B under a change of metric The operator Γ g does not change under the change of metric. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
43 Π B under a change of metric The operator Γ g does not change under the change of metric. However, Proposition Γ g = ΘΓ g C where Θ is a bounded multiplication operator on L2 (M) and C is a bounded multiplication operator on L 2 (M) L 2 (T M). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
44 Π B under a change of metric The operator Γ g does not change under the change of metric. However, Proposition Γ g = ΘΓ g C where Θ is a bounded multiplication operator on L2 (M) and C is a bounded multiplication operator on L 2 (M) L 2 (T M). Thus, Π B,g = Γ g + B 1 Γ gb 2 = Γ g + B 1 ΘΓ gcb 2. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
45 Π B under a change of metric The operator Γ g does not change under the change of metric. However, Proposition Γ g = ΘΓ g C where Θ is a bounded multiplication operator on L2 (M) and C is a bounded multiplication operator on L 2 (M) L 2 (T M). Thus, Π B,g = Γ g + B 1 Γ gb 2 = Γ g + B 1 ΘΓ gcb 2. The idea is to reduce the Kato problem for Π B,g to a Kato problem for Π B, g = Γ g + B 1 Γ g B 2 where B 1 = B 1 Θ and B 2 = CB 2 but now with a smooth metric g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
46 Ellipticity Let us define ellipticity of L A with respect to a metric h by Re av, v h κ 1,h v 2 2,h, v L2 (M) Re ASu, Su h κ 2,h u 2 H 1,h, u H1 (M). (E) Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
47 Ellipticity (cont.) Then the loss in the ellipticity constants by transferring from one operator to another is as follows. Proposition If g g op,g δ < 1, then assuming (E) with respect to g implies (E) with respect to g with an appropriate change of the coefficients a and A with ellipticity constants κ 1, g = κ 1,g (1 + δ) n 2 (1 δ) and κ 2, g = κ 2,g. (1 + δ) n 2 Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
48 Theorem (Reduction of the rough problem to the smooth) Let M be a smooth manifold with a C 0 metric g and suppose that the ellipticity condition (E) is satisfied for (M, g). Suppose further that there exists a smooth, complete metric g satisfying: (i) there exists η > 0 such that Ric g g η, (ii) there exists κ > 0 such that inj(m, g) κ, (iii) g g op,g < 1, Then, the quadratic estimate ˆ holds for all u R(Π B,g ). 0 tπ B,g (I + t 2 Π 2 B,g) 1 u 2 g dt t u 2 2,g Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
49 Compact manifolds Given a C 0 metric g, we can always find a C metric g that is as close as we would like in the op,s,g norm by pasting together Euclidean metrics via a partition of unity. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
50 Compact manifolds Given a C 0 metric g, we can always find a C metric g that is as close as we would like in the op,s,g norm by pasting together Euclidean metrics via a partition of unity. Further if we assume that M is compact, then automatically Ric g C g and inj(m, g) κ g > 0. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
51 Compact manifolds Given a C 0 metric g, we can always find a C metric g that is as close as we would like in the op,s,g norm by pasting together Euclidean metrics via a partition of unity. Further if we assume that M is compact, then automatically Ric g C g and inj(m, g) κ g > 0. Theorem Let M be a smooth, compact Riemannian manifold and let g be a C 0 metric on M. Then, the quadratic estimate ˆ 0 tπb,g (I + tπ 2 B,g) 1 u 2 dt t u 2 is satisfied for all u R(Π B,g ). Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
52 The noncompact case Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
53 The noncompact case Although we can obtain arbitrarily close smooth metrics via a partition of unity in the noncompact setting, the desired geometric properties are not automatic. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
54 The noncompact case Although we can obtain arbitrarily close smooth metrics via a partition of unity in the noncompact setting, the desired geometric properties are not automatic. Intuition is that the smoothing properties of geometric flows could be utilised to find good close smooth metrics. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
55 The noncompact case Although we can obtain arbitrarily close smooth metrics via a partition of unity in the noncompact setting, the desired geometric properties are not automatic. Intuition is that the smoothing properties of geometric flows could be utilised to find good close smooth metrics. The non-uniqueness of existence of solutions to geometric flows are not an issue since we only require one good metric near the initial one. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
56 The noncompact case Although we can obtain arbitrarily close smooth metrics via a partition of unity in the noncompact setting, the desired geometric properties are not automatic. Intuition is that the smoothing properties of geometric flows could be utilised to find good close smooth metrics. The non-uniqueness of existence of solutions to geometric flows are not an issue since we only require one good metric near the initial one. A good place to start may be the mean curvature flow since such flows have been studied with rough initial data. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
57 The noncompact case Although we can obtain arbitrarily close smooth metrics via a partition of unity in the noncompact setting, the desired geometric properties are not automatic. Intuition is that the smoothing properties of geometric flows could be utilised to find good close smooth metrics. The non-uniqueness of existence of solutions to geometric flows are not an issue since we only require one good metric near the initial one. A good place to start may be the mean curvature flow since such flows have been studied with rough initial data. The first task is to understand the backward behaviour of this flow and the relationship it has to op,g. Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
58 References I [AHLMcT] Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on R n, Ann. of Math. (2) 156 (2002), no. 2, [AKMc] Andreas Axelsson, Stephen Keith, and Alan McIntosh, Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (2006), no. 3, [BMc] L. Bandara and A. McIntosh, The Kato square root problem on vector bundles with generalised bounded geometry, ArXiv e-prints (2012). [Kato61] Tosio Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan 13 (1961), MR (25 #1453) [Mc72] Alan McIntosh, On the comparability of A 1/2 and A 1/2, Proc. Amer. Math. Soc. 32 (1972), MR (44 #7354) Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
59 References II [Mc82], On representing closed accretive sesquilinear forms as (A 1/2 u, A 1/2 v), Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. III (Paris, 1980/1981), Res. Notes in Math., vol. 70, Pitman, Boston, Mass., 1982, pp MR (84k:47030) Lashi Bandara (ANU) L operators and non-smooth metrics 17 October / 23
The Kato square root problem on vector bundles with generalised bounded geometry
The Kato square root problem on vector bundles with generalised bounded geometry Lashi Bandara (Joint work with Alan McIntosh, ANU) Centre for Mathematics and its Applications Australian National University
More informationThe Kato square root problem on vector bundles with generalised bounded geometry
The Kato square root problem on vector bundles with generalised bounded geometry Lashi Bandara (Joint work with Alan McIntosh, ANU) Centre for Mathematics and its Applications Australian National University
More informationGeometry and the Kato square root problem
Geometry and the Kato square root problem Lashi Bandara Centre for Mathematics and its Applications Australian National University 29 July 2014 Geometric Analysis Seminar Beijing International Center for
More informationGeometry and the Kato square root problem
Geometry and the Kato square root problem Lashi Bandara Centre for Mathematics and its Applications Australian National University 7 June 2013 Geometric Analysis Seminar University of Wollongong Lashi
More informationGeometry and the Kato square root problem
Geometry and the Kato square root problem Lashi Bandara Centre for Mathematics and its Applications Australian National University 7 June 2013 Geometric Analysis Seminar University of Wollongong Lashi
More informationRough metrics, the Kato square root problem, and the continuity of a flow tangent to the Ricci flow
Rough metrics, the Kato square root problem, and the continuity of a flow tangent to the Ricci flow Lashi Bandara Mathematical Sciences Chalmers University of Technology and University of Gothenburg 22
More informationSquare roots of perturbed sub-elliptic operators on Lie groups
Square roots of perturbed sub-elliptic operators on Lie groups Lashi Bandara (Joint work with Tom ter Elst, Auckland and Alan McIntosh, ANU) Centre for Mathematics and its Applications Australian National
More informationSquare roots of operators associated with perturbed sub-laplacians on Lie groups
Square roots of operators associated with perturbed sub-laplacians on Lie groups Lashi Bandara maths.anu.edu.au/~bandara (Joint work with Tom ter Elst, Auckland and Alan McIntosh, ANU) Mathematical Sciences
More informationQuadratic estimates and perturbations of Dirac type operators on doubling measure metric spaces
Quadratic estimates and perturbations of Dirac type operators on doubling measure metric spaces Lashi Bandara maths.anu.edu.au/~bandara Mathematical Sciences Institute Australian National University February
More informationLocal Hardy Spaces and Quadratic Estimates for Dirac Type Operators on Riemannian Manifolds
Local Hardy Spaces and Quadratic Estimates for Dirac Type Operators on Riemannian Manifolds Andrew J. Morris June 2010 A thesis submitted for the degree of Doctor of Philosophy. of The Australian National
More informationCONTINUITY OF SOLUTIONS TO SPACE-VARYING POINTWISE LINEAR ELLIPTIC EQUATIONS. Lashi Bandara
Publ. Mat. 61 (2017), 239 258 DOI: 10.5565/PUBLMAT 61117 09 CONTINUITY OF SOLUTIONS TO SPACE-VARYING POINTWISE LINEAR ELLIPTIC EQUATIONS Lashi Bandara Abstract: We consider pointwise linear elliptic equations
More informationarxiv: v1 [math.dg] 31 Dec 2018
FUNCTIONAL CALCULUS AND HARMONIC ANALYSIS IN GEOMETRY LASHI BANDARA arxiv:1812.11795v1 [math.dg] 31 Dec 218 Abstract. In this short survey article, we showcase a number of non-trivial geometric problems
More informationSELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
More informationTHE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS
THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily
More informationTitle: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on
Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on manifolds. Author: Ognjen Milatovic Department Address: Department
More informationHardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus.
Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus. Xuan Thinh Duong (Macquarie University, Australia) Joint work with Ji Li, Zhongshan
More informationIs a dual mesh really necessary?
Is a dual mesh really necessary? Paul Leopardi Mathematical Sciences Institute, Australian National University. For presentation at ICIAM, Vancouver, 2011. Joint work with Ari Stern, UCSD. 19 July 2011
More informationLecture 4: Harmonic forms
Lecture 4: Harmonic forms Jonathan Evans 29th September 2010 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 1 / 15 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 2 / 15
More informationL19: Fredholm theory. where E u = u T X and J u = Formally, J-holomorphic curves are just 1
L19: Fredholm theory We want to understand how to make moduli spaces of J-holomorphic curves, and once we have them, how to make them smooth and compute their dimensions. Fix (X, ω, J) and, for definiteness,
More informationNotes by Maksim Maydanskiy.
SPECTRAL FLOW IN MORSE THEORY. 1 Introduction Notes by Maksim Maydanskiy. Spectral flow is a general formula or computing the Fredholm index of an operator d ds +A(s) : L1,2 (R, H) L 2 (R, H) for a family
More informationThe heat kernel meets Approximation theory. theory in Dirichlet spaces
The heat kernel meets Approximation theory in Dirichlet spaces University of South Carolina with Thierry Coulhon and Gerard Kerkyacharian Paris - June, 2012 Outline 1. Motivation and objectives 2. The
More informationarxiv: v1 [math.ap] 18 May 2017
Littlewood-Paley-Stein functions for Schrödinger operators arxiv:175.6794v1 [math.ap] 18 May 217 El Maati Ouhabaz Dedicated to the memory of Abdelghani Bellouquid (2/2/1966 8/31/215) Abstract We study
More informationCalderón-Zygmund inequality on noncompact Riem. manifolds
The Calderón-Zygmund inequality on noncompact Riemannian manifolds Institut für Mathematik Humboldt-Universität zu Berlin Geometric Structures and Spectral Invariants Berlin, May 16, 2014 This talk is
More informationA Remark on -harmonic Functions on Riemannian Manifolds
Electronic Journal of ifferential Equations Vol. 1995(1995), No. 07, pp. 1-10. Published June 15, 1995. ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110
More informationFunctions with bounded variation on Riemannian manifolds with Ricci curvature unbounded from below
Functions with bounded variation on Riemannian manifolds with Ricci curvature unbounded from below Institut für Mathematik Humboldt-Universität zu Berlin ProbaGeo 2013 Luxembourg, October 30, 2013 This
More informationOn m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry
On m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry Ognjen Milatovic Department of Mathematics and Statistics University of North Florida Jacksonville, FL 32224 USA. Abstract
More informationQuantising proper actions on Spin c -manifolds
Quantising proper actions on Spin c -manifolds Peter Hochs University of Adelaide Differential geometry seminar Adelaide, 31 July 2015 Joint work with Mathai Varghese (symplectic case) Geometric quantization
More informationLocalizing solutions of the Einstein equations
Localizing solutions of the Einstein equations Richard Schoen UC, Irvine and Stanford University - General Relativity: A Celebration of the 100th Anniversary, IHP - November 20, 2015 Plan of Lecture The
More informationDevoir 1. Valerio Proietti. November 5, 2014 EXERCISE 1. z f = g. supp g EXERCISE 2
GÉOMÉTRIE ANALITIQUE COMPLEXE UNIVERSITÉ PARIS-SUD Devoir Valerio Proietti November 5, 04 EXERCISE Thanks to Corollary 3.4 of [, p. ] we know that /πz is a fundamental solution of the operator z on C.
More informationWEIGHTED NORM INEQUALITIES, OFF-DIAGONAL ESTIMATES AND ELLIPTIC OPERATORS PASCAL AUSCHER AND JOSÉ MARÍA MARTELL
Proceedings of the 8th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 16-2, 28 Contemporary Mathematics 55 21), 61--83 WEIGHTED NORM INEQUALITIES, OFF-DIAGONAL
More informationESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS. Zhongwei Shen
W,p ESTIMATES FOR ELLIPTIC HOMOGENIZATION PROBLEMS IN NONSMOOTH DOMAINS Zhongwei Shen Abstract. Let L = div`a` x, > be a family of second order elliptic operators with real, symmetric coefficients on a
More informationarxiv: v1 [math.dg] 18 Aug 2016
EVOLUTION OF AREA-DECREASING MAPS BETWEEN TWO-DIMENSIONAL EUCLIDEAN SPACES FELIX LUBBE arxiv:608.05394v [math.dg] 8 Aug 206 Abstract. We consider the mean curvature flow of the graph of a smooth map f
More informationOptimal Transportation. Nonlinear Partial Differential Equations
Optimal Transportation and Nonlinear Partial Differential Equations Neil S. Trudinger Centre of Mathematics and its Applications Australian National University 26th Brazilian Mathematical Colloquium 2007
More informationMotivation towards Clifford Analysis: The classical Cauchy s Integral Theorem from a Clifford perspective
Motivation towards Clifford Analysis: The classical Cauchy s Integral Theorem from a Clifford perspective Lashi Bandara November 26, 29 Abstract Clifford Algebras generalise complex variables algebraically
More informationReconstruction theorems in noncommutative Riemannian geometry
Reconstruction theorems in noncommutative Riemannian geometry Department of Mathematics California Institute of Technology West Coast Operator Algebra Seminar 2012 October 21, 2012 References Published
More informationOpen Questions in Quantum Information Geometry
Open Questions in Quantum Information Geometry M. R. Grasselli Department of Mathematics and Statistics McMaster University Imperial College, June 15, 2005 1. Classical Parametric Information Geometry
More informationOn non negative solutions of some quasilinear elliptic inequalities
On non negative solutions of some quasilinear elliptic inequalities Lorenzo D Ambrosio and Enzo Mitidieri September 28 2006 Abstract Let f : R R be a continuous function. We prove that under some additional
More informationAn abstract Hodge Dirac operator and its stable discretization
An abstract Hodge Dirac operator and its stable discretization Paul Leopardi Mathematical Sciences Institute, Australian National University. For presentation at ICCA, Tartu,. Joint work with Ari Stern,
More informationRIGIDITY OF STATIONARY BLACK HOLES WITH SMALL ANGULAR MOMENTUM ON THE HORIZON
RIGIDITY OF STATIONARY BLACK HOLES WITH SMALL ANGULAR MOMENTUM ON THE HORIZON S. ALEXAKIS, A. D. IONESCU, AND S. KLAINERMAN Abstract. We prove a black hole rigidity result for slowly rotating stationary
More informationMATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY
MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY 0.1. Vector Bundles and Connection 1-forms. Let E X be a complex vector bundle of rank r over a smooth manifold. Recall the following abstract
More informationTHE FUNDAMENTAL GROUP OF NON-NEGATIVELY CURVED MANIFOLDS David Wraith The aim of this article is to oer a brief survey of an interesting, yet accessib
THE FUNDAMENTAL GROUP OF NON-NEGATIVELY CURVED MANIFOLDS David Wraith The aim of this article is to oer a brief survey of an interesting, yet accessible line of research in Dierential Geometry. A fundamental
More informationOn the Convergence of a Modified Kähler-Ricci Flow. 1 Introduction. Yuan Yuan
On the Convergence of a Modified Kähler-Ricci Flow Yuan Yuan Abstract We study the convergence of a modified Kähler-Ricci flow defined by Zhou Zhang. We show that the modified Kähler-Ricci flow converges
More informationMetric Structures for Riemannian and Non-Riemannian Spaces
Misha Gromov with Appendices by M. Katz, P. Pansu, and S. Semmes Metric Structures for Riemannian and Non-Riemannian Spaces Based on Structures Metriques des Varietes Riemanniennes Edited by J. LaFontaine
More informationMinimization problems on the Hardy-Sobolev inequality
manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev
More informationDifferential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18
Differential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18 Required problems (to be handed in): 2bc, 3, 5c, 5d(i). In doing any of these problems, you may assume the results
More informationA REMARK ON AN EQUATION OF WAVE MAPS TYPE WITH VARIABLE COEFFICIENTS
A REMARK ON AN EQUATION OF WAVE MAPS TYPE WITH VARIABLE COEFFICIENTS DAN-ANDREI GEBA Abstract. We obtain a sharp local well-posedness result for an equation of wave maps type with variable coefficients.
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationNOTE ON ASYMPTOTICALLY CONICAL EXPANDING RICCI SOLITONS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 NOTE ON ASYMPTOTICALLY CONICAL EXPANDING RICCI SOLITONS JOHN LOTT AND PATRICK WILSON (Communicated
More informationGeneralized Schrödinger semigroups on infinite weighted graphs
Generalized Schrödinger semigroups on infinite weighted graphs Institut für Mathematik Humboldt-Universität zu Berlin QMATH 12 Berlin, September 11, 2013 B.G, Ognjen Milatovic, Francoise Truc: Generalized
More informationPractice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009.
Practice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009. Solutions (1) Let Γ be a discrete group acting on a manifold M. (a) Define what it means for Γ to act freely. Solution: Γ acts
More informationHodge de Rham decomposition for an L 2 space of differfential 2-forms on path spaces
Hodge de Rham decomposition for an L 2 space of differfential 2-forms on path spaces K. D. Elworthy and Xue-Mei Li For a compact Riemannian manifold the space L 2 A of L 2 differential forms decomposes
More informationStrictly convex functions on complete Finsler manifolds
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 4, November 2016, pp. 623 627. DOI 10.1007/s12044-016-0307-2 Strictly convex functions on complete Finsler manifolds YOE ITOKAWA 1, KATSUHIRO SHIOHAMA
More informationIsometries of Riemannian and sub-riemannian structures on 3D Lie groups
Isometries of Riemannian and sub-riemannian structures on 3D Lie groups Rory Biggs Geometry, Graphs and Control (GGC) Research Group Department of Mathematics, Rhodes University, Grahamstown, South Africa
More informationThe Cheeger-Müller theorem and generalizations
Presentation Universidade Federal de São Carlos - UFSCar February 21, 2013 This work was partially supported by Grant FAPESP 2008/57607-6. FSU Topology Week Presentation 1 Reidemeister Torsion 2 3 4 Generalizations
More informationThe Hopf argument. Yves Coudene. IRMAR, Université Rennes 1, campus beaulieu, bat Rennes cedex, France
The Hopf argument Yves Coudene IRMAR, Université Rennes, campus beaulieu, bat.23 35042 Rennes cedex, France yves.coudene@univ-rennes.fr slightly updated from the published version in Journal of Modern
More informationQuantising noncompact Spin c -manifolds
Quantising noncompact Spin c -manifolds Peter Hochs University of Adelaide Workshop on Positive Curvature and Index Theory National University of Singapore, 20 November 2014 Peter Hochs (UoA) Noncompact
More informationBERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS
BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS SHOO SETO Abstract. These are the notes to an expository talk I plan to give at MGSC on Kähler Geometry aimed for beginning graduate students in hopes to motivate
More informationHolonomy groups. Thomas Leistner. Mathematics Colloquium School of Mathematics and Physics The University of Queensland. October 31, 2011 May 28, 2012
Holonomy groups Thomas Leistner Mathematics Colloquium School of Mathematics and Physics The University of Queensland October 31, 2011 May 28, 2012 1/17 The notion of holonomy groups is based on Parallel
More informationNote on the Chen-Lin Result with the Li-Zhang Method
J. Math. Sci. Univ. Tokyo 18 (2011), 429 439. Note on the Chen-Lin Result with the Li-Zhang Method By Samy Skander Bahoura Abstract. We give a new proof of the Chen-Lin result with the method of moving
More informationChanging sign solutions for the CR-Yamabe equation
Changing sign solutions for the CR-Yamabe equation Ali Maalaoui (1) & Vittorio Martino (2) Abstract In this paper we prove that the CR-Yamabe equation on the Heisenberg group has infinitely many changing
More informationTHE UNIFORMISATION THEOREM OF RIEMANN SURFACES
THE UNIFORISATION THEORE OF RIEANN SURFACES 1. What is the aim of this seminar? Recall that a compact oriented surface is a g -holed object. (Classification of surfaces.) It can be obtained through a 4g
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationRecent developments in elliptic partial differential equations of Monge Ampère type
Recent developments in elliptic partial differential equations of Monge Ampère type Neil S. Trudinger Abstract. In conjunction with applications to optimal transportation and conformal geometry, there
More informationMoment Maps and Toric Special Holonomy
Department of Mathematics, University of Aarhus PADGE, August 2017, Leuven DFF - 6108-00358 Delzant HyperKähler G 2 Outline The Delzant picture HyperKähler manifolds Hypertoric Construction G 2 manifolds
More informationarxiv: v2 [math.dg] 23 Mar 2016
arxiv:1508.03417v2 [math.dg] 23 Mar 2016 AN EXISTENCE TIME ESTIMATE FOR KÄHLER-RICCI FLOW ALBERT CHAU 1, KA-FAI LI, AND LUEN-FAI TAM 2 Abstract. FixacompletenoncompactKählermanifold(M n,h 0 ) with bounded
More informationConstructing compact 8-manifolds with holonomy Spin(7)
Constructing compact 8-manifolds with holonomy Spin(7) Dominic Joyce, Oxford University Simons Collaboration meeting, Imperial College, June 2017. Based on Invent. math. 123 (1996), 507 552; J. Diff. Geom.
More informationBloch radius, normal families and quasiregular mappings
Bloch radius, normal families and quasiregular mappings Alexandre Eremenko Abstract Bloch s Theorem is extended to K-quasiregular maps R n S n, where S n is the standard n-dimensional sphere. An example
More informationOrigin, Development, and Dissemination of Differential Geometry in Mathema
Origin, Development, and Dissemination of Differential Geometry in Mathematical History The Borough of Manhattan Community College -The City University of New York Fall 2016 Meeting of the Americas Section
More information1 k x k. d(x, y) =sup k. y k = max
1 Lecture 13: October 8 Urysohn s metrization theorem. Today, I want to explain some applications of Urysohn s lemma. The first one has to do with the problem of characterizing metric spaces among all
More informationPseudo-Poincaré Inequalities and Applications to Sobolev Inequalities
Pseudo-Poincaré Inequalities and Applications to Sobolev Inequalities Laurent Saloff-Coste Abstract Most smoothing procedures are via averaging. Pseudo-Poincaré inequalities give a basic L p -norm control
More informationHyperbolic Geometric Flow
Hyperbolic Geometric Flow Kefeng Liu Zhejiang University UCLA Page 1 of 41 Outline Introduction Hyperbolic geometric flow Local existence and nonlinear stability Wave character of metrics and curvatures
More informationThe p-laplacian and geometric structure of the Riemannian manifold
Workshop on Partial Differential Equation and its Applications Institute for Mathematical Sciences - National University of Singapore The p-laplacian and geometric structure of the Riemannian manifold
More informationNorms of Elementary Operators
Irish Math. Soc. Bulletin 46 (2001), 13 17 13 Norms of Elementary Operators RICHARD M. TIMONEY Abstract. We make some simple remarks about the norm problem for elementary operators in the contexts of algebras
More informationINSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD
INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:
More informationQuasi-invariant measures on the path space of a diffusion
Quasi-invariant measures on the path space of a diffusion Denis Bell 1 Department of Mathematics, University of North Florida, 4567 St. Johns Bluff Road South, Jacksonville, FL 32224, U. S. A. email: dbell@unf.edu,
More informationA note on the Stokes operator and its powers
J Appl Math Comput (2011) 36: 241 250 DOI 10.1007/s12190-010-0400-0 JAMC A note on the Stokes operator and its powers Jean-Luc Guermond Abner Salgado Received: 3 December 2009 / Published online: 28 April
More informationCitation Osaka Journal of Mathematics. 41(4)
TitleA non quasi-invariance of the Brown Authors Sadasue, Gaku Citation Osaka Journal of Mathematics. 414 Issue 4-1 Date Text Version publisher URL http://hdl.handle.net/1194/1174 DOI Rights Osaka University
More informationOverview of normed linear spaces
20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural
More informationRIEMANNIAN SUBMERSIONS NEED NOT PRESERVE POSITIVE RICCI CURVATURE
RIEMANNIAN SUBMERSIONS NEED NOT PRESERVE POSITIVE RICCI CURVATURE CURTIS PRO AND FREDERICK WILHELM Abstract. If π : M B is a Riemannian Submersion and M has positive sectional curvature, O Neill s Horizontal
More informationA SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY
A SHARP STABILITY ESTIMATE IN TENSOR TOMOGRAPHY PLAMEN STEFANOV 1. Introduction Let (M, g) be a compact Riemannian manifold with boundary. The geodesic ray transform I of symmetric 2-tensor fields f is
More informationthe neumann-cheeger constant of the jungle gym
the neumann-cheeger constant of the jungle gym Itai Benjamini Isaac Chavel Edgar A. Feldman Our jungle gyms are dimensional differentiable manifolds M, with preferred Riemannian metrics, associated to
More informationLeft-invariant Einstein metrics
on Lie groups August 28, 2012 Differential Geometry seminar Department of Mathematics The Ohio State University these notes are posted at http://www.math.ohio-state.edu/ derdzinski.1/beamer/linv.pdf LEFT-INVARIANT
More informationSpectral Triples on the Sierpinski Gasket
Spectral Triples on the Sierpinski Gasket Fabio Cipriani Dipartimento di Matematica Politecnico di Milano - Italy ( Joint works with D. Guido, T. Isola, J.-L. Sauvageot ) AMS Meeting "Analysis, Probability
More information1 Continuity Classes C m (Ω)
0.1 Norms 0.1 Norms A norm on a linear space X is a function : X R with the properties: Positive Definite: x 0 x X (nonnegative) x = 0 x = 0 (strictly positive) λx = λ x x X, λ C(homogeneous) x + y x +
More informationVelocity averaging a general framework
Outline Velocity averaging a general framework Martin Lazar BCAM ERC-NUMERIWAVES Seminar May 15, 2013 Joint work with D. Mitrović, University of Montenegro, Montenegro Outline Outline 1 2 L p, p >= 2 setting
More informationAbundance of stable ergodicity
Abundance of stable ergodicity Christian Bonatti, Carlos Matheus, Marcelo Viana, Amie Wilkinson October 5, 2004 Abstract We consider the set PH ω (M) of volume preserving partially hyperbolic diffeomorphisms
More informationEXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018
EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018 While these notes are under construction, I expect there will be many typos. The main reference for this is volume 1 of Hörmander, The analysis of liner
More informationA class of non-convex polytopes that admit no orthonormal basis of exponentials
A class of non-convex polytopes that admit no orthonormal basis of exponentials Mihail N. Kolountzakis and Michael Papadimitrakis 1 Abstract A conjecture of Fuglede states that a bounded measurable set
More informationSchrodinger operator: optimal decay of fundamental solutions and boundary value problems
Schrodinger operator: optimal decay of fundamental solutions and boundary value problems University of Minnesota Department of Mathematics PCMI Research Program Seminar Park City, Utah July 17, 2018 1
More informationarxiv: v1 [math-ph] 18 Feb 2015
ON THE THEORY OF SELF-ADJOINT EXTENSIONS OF SYMMETRIC OPERATORS AND ITS APPLICATIONS TO QUANTUM PHYSICS A. IBORT AND J.M. PÉREZ-PARDO arxiv:1502.05229v1 [math-ph] 18 Feb 2015 Abstract. This is a series
More informationBIHARMONIC WAVE MAPS INTO SPHERES
BIHARMONIC WAVE MAPS INTO SPHERES SEBASTIAN HERR, TOBIAS LAMM, AND ROLAND SCHNAUBELT Abstract. A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed.
More informationThe spectral zeta function
The spectral zeta function Bernd Ammann June 4, 215 Abstract In this talk we introduce spectral zeta functions. The spectral zeta function of the Laplace-Beltrami operator was already introduced by Minakshisundaram
More information1.4 The Jacobian of a map
1.4 The Jacobian of a map Derivative of a differentiable map Let F : M n N m be a differentiable map between two C 1 manifolds. Given a point p M we define the derivative of F at p by df p df (p) : T p
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Harmonic spinors and local deformations of the metric Bernd Ammann, Mattias Dahl, and Emmanuel Humbert Preprint Nr. 03/2010 HARMONIC SPINORS AND LOCAL DEFORMATIONS OF
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationCitation Osaka Journal of Mathematics. 43(1)
TitleA note on compact solvmanifolds wit Author(s) Hasegawa, Keizo Citation Osaka Journal of Mathematics. 43(1) Issue 2006-03 Date Text Version publisher URL http://hdl.handle.net/11094/11990 DOI Rights
More informationSpacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds
Spacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds Alfonso Romero Departamento de Geometría y Topología Universidad de Granada 18071-Granada Web: http://www.ugr.es/
More informationClarkson Inequalities With Several Operators
isid/ms/2003/23 August 14, 2003 http://www.isid.ac.in/ statmath/eprints Clarkson Inequalities With Several Operators Rajendra Bhatia Fuad Kittaneh Indian Statistical Institute, Delhi Centre 7, SJSS Marg,
More informationL 2 Geometry of the Symplectomorphism Group
University of Notre Dame Workshop on Innite Dimensional Geometry, Vienna 2015 Outline 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of Outline 1 The Exponential Map on D s ω(m) 2 Existence
More information